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ELEMENTS 


DIFFERENTIAL  AND   INTEGRAL 
CALCULUS. 

METHOD    OF   RATES. 


BY 

ARTHUR    SHERBURNE    HARDY,    I'H.D., 

Professor  of  Mathematics  in  Dartmouth  College. 


UNJVfcft&i 


.  BOSTON,    U.S.A.: 
PUBLISHED  BY   GINN  &  COMPANY. 
1890. 


Copyright,  1890, 
By  ARTHUR  SHERBURNE  HARDY, 


All  Rights  Reserved. 


Ttpoorapht  bt  J.  S.  CusHiNG  &  Co.,  Boston,  U.S.A. 


Pbesswobk  by  Ginn  &  Co.,  Boston,  U.S.A. 


PEEFACE. 


This  text-book  is  based  on  the  method  of  rates,  which,  in 
the  experience  of  the  author,  has  proved  most  satisfactory 
in  a  first  presentation  of  the  object  and  scope  of  the  Calculus. 
No  comparisons  have  been  made  between  this  method  and 
those  of  limits  or  of  infinitesimals.  This  larger  view  of  the 
Calculus,  and  of  mathematical  reasoning  and  processes  in 
general,  cannot  readily  be  given  with  good  results  in  the 
brief  time  allotted  the  subject  in  the  general  college  course. 

The  immediate  object  of  the  Differential  Calculus  is  the 
measurement  and  comparison  of  rates  of  change  when  the 
change  is  not  uniform.  Whether  a  quantity  is  or  is  not 
changing  uniformly,  however,  the  rate  at  any  instant  is  de- 
termined in  essentially  the  same  manner ;  viz.  by  ascertaining 
what  its  change  would  have  been  in  a  unit  of  time  had  its 
rate  remained  what  it  was  at  the  instant  in  question.  It  is 
this  change  which  the  Calcidus  enables  us  to  determine, 
however  complicated  the  law  of  variation  may  be.  This 
conception  of  the  nature  of  the  problem  is  simple,  and  seems 
to  afford  the  best  foundation  for  further  and  more  compre- 
hensive study;    while  for   those    who   are   not   to    make   a 


183665 


IV  PREFACE. 

special  study  of  mathematics  it   secures   a  more  intelligent 

and  less   mechanical   grasp   of  the   problems   involved  than 

other  methods  whose  conceptions   and  logic  are   not  easily 

mastered  in  undergraduate  courses. 

My  thanks  are  due  to  Professor  Worthen,  my  colleague, 

for  valuable   suggestions   and   assistance   in   the   reading   of 

proofs. 

ARTHUR    SHERBURNE   HARDY. 

Hamoveb,  N.H.,  June  2,  1890. 


CONTENTS. 


Part  I.  —  The  Differential  Calculus. 


.    chapter  I.  —  introductory  theorems. 

ART.  PAGE 

1.  Quantities  of  the  Calculus 8 

2.  Functions :5 

3.  Classification  of  functions 4 

4.  Increments 6 

5.  Uniform  change 6 

6.  Uniform  motion 6 

7.  Varied  change 7 

8.  Differentials 8 

9.  Distinctions  between  increment,  differential,  and  rate n      __  8 

10.  Corresponding  differentials  and  simultaneous  rates . . .  -p.,.,-.^ 9 

11.  Symbol  of  a  rate 9 

12.  Corresponding  differentials  of  equals  are  equal 9 

13.  Object  of  the  Differential  Calculus 10 

14.  Differentiation 11 


chapter  II.  —  differentiation  op  explicit  functions. 

The  Algebraic  Functions. 

15.  Differential  of  a  constant 12 

10.  Differential  of  x  +  z-v 12 

17.  Differential  of  mx 12 

18.  Differential  of  xz 13 

19.  Differential  of  xzv 13 

20.  Differential  of  - 13 

z 

21.  Differential  of  ar».     Examples 14 


VI  CONTENTS. 

ART.  PAGE 

22.  Analytic  signification  of  y- 18 

23.  Applications 19 

24.  Geometric  signification  of  -7^ 24 

25.  Relations  between  — ,   — ,   ^ 25 

dt     dt     (It 

20.  Expressions  for  ~  and  -^ 26 

(Is  (Is 

27.  Applications 26 


The  Transcendental  Functions. 

The  Logarithmic  and  Expoiietitial  Funrtio)is. 

28.  Differential  of  log.r 31 

29.  Differential  of  a*.     Examples 32 

30.  Applications 36 

The  Trigonometric  Functions. 

31.  Circular  measure  of  an  angle 37 

32.  Differential  of  sinr '. 38 

33.  Differential  of  cos  a- 38 

34.  Differential  of  tan  x 39 

35.  Differential  of  cot  x 39 

36.  Differential  of  sec  x 39 

37.  Differential  of  cosecx 39 

38.  Differential  of  vers  x 40 

39.  Differential  of  covers  .r.     Examples 40 

The  Circular  Functions. 

40.  Differential  of  sin-'  x 41 

41 .  Differential  of  cos-i  x 41 

42.  Differential  of  tan  1  x 42 

43.  Differential  of  cot-'  x 42 

44.  Differential  of  sec  -'  t 42 

45.  Differential  of  cosec  1  .r 42 

46.  Differential  of  vers  '  x 43 

47.  Differential  of  covers- '  x.     Examples 43 

48.  Applications 45 


( 


CONTENTS.  Vll 


CHAPTER   III.  —  SUCCESSIVE  DIFFERENTIATION. 

ART.  PAGE 

49.  Equicrescent  variable 49 

50.  Differential  of  an  equicrescent  variable 49 

51.  Successive  derived  equations 49 

52.  Notation 50 

53.  Remark  on  the  equicrescent  variable.     Examples 50 

54.  Successive  derivatives 52 

55.  Sign  of  the  nth  derivative.     Examples 54 

56.  Derived  functions  which  become  oo 66 

67.  Notation 67 

68.  Change  of  the  equicrescent  variable.    Examples 57 

Applications  of  Successive  Differentiation. 

Accelei-ations. 

59.  Accelerations 60 

60.  Signs  of  the  axial  accelerations.    Examples 61 

Development  of  Continuous  Functions. 

61.  Limit  of  a  variable 63 

62.  Geometrical  limit 64 

63.  Two  meanings  of  limit 64 

64.  A  quantity  cannot  have  two  limits 65 

65.  Continuous  functions 65 

66.  Series 65 

67.  Sum  of  a  series 65 

68.  Development  of  a  function 65 

69.  Maclaurin's  theorem ". 66 

70.  Taylor's  theorem 68 

71.  Completion  of  Taylor's  and  Maclaurin's  formulaj 69 

72.  Applications 72 

73.  Failing  cases  of  Taylor's  and  Maclaurin's  formulai 80 

Evaluation  of  lUusorij  Forms. 

74.  The  form  -•     Examples 81 

75.  The  form  — •     Examples 84 

76.  Tlie  form  0 x «.     Examples 87 

77.  The  form  oo  —  oo .     Examples 88 

78.  The  forms  00 0,  1*,  00.    Examples 89 


VIU  CONTENTS. 

Maxima  and  Minima. 

ART.  PAGE 

79.  Definition  of  maxima  and  minima  values  91 

80.  Condition  of  a  maximum  or  minimum  value 91 

81.  Geometric  illustrations 92 

82.  Examination  of  the  critical  values  when  /'(a-)  =  0 93 

83.  General  method 93 

84.  Abbreviated  processes.    Examples 96 

85.  Examination  of  the  critical  values  when /'(a-)  =  00.    Examples..  99 

86.  Geometrical  problems 100 


CHAPTEB  IV.  —  FUNCTIONS  OF  TWO  OR   MORE  VARIABLES. 

87.  Partial  differentials Ill 

88.  Notation T Ill 

89.  Partial  derivatives.     Examples Ill 

90.  Total  differential.    Examples 113 

91.  Total  derivative 114 

92.  Total  derivative  with  respect  to  any  variable.     Examples 115 

93.  Implicit  functions  of  two  variables.    Examples 117 

94.  Evaluation  of  the  first  derivative  of  an  implicit  function.    P]x- 

amples 118 


98.   ~  in  terms  of  x  and  y 124 


CHAPTER  v.  — PLANE  CURVES. 
Curvature. 

95.  Direction  of  curvature 121 

9G.  Points  of  inflexion.    Examples 121 

97.  Rate  of  curvature 123 

ds 

99.  Curvature  of  the  circle 125 

100.  Radius  of  curvature 125 

101 .  Centre  of  curvature 126 

102.  Maximum  or  minimum  curvature 127 

103.  Intersection  of  the  curve  and  circle  of  curvature.    Examples...  127 

Evolntes   and  £nTelopes. 

104.  Evolute  and  involute 129 

105.  Equation  of  the  evolute.     Examples 129 


CONTENTS.  ix 

ART.  PAOI 

106.  Envelopes 132 

107.  Equation  of  the  envelope.    Examples 133 

108.  The  evolute  is  the  envelope  of  the  normals  to  the  involute 137 

109.  Property  of  the  involute  and  evolute 138 

110.  Mechanical  construction  of  the  involute 138 

111.  Orders  of  contact.    Examples 138 

^ing^ular  Points. 

112.  Multiple,  cu.sp,  and  salient  points 140 

113.  Conjugate  points 141 

114.  Determination  of  singular  points  by  inspection.     Examples. . . .  141 

Asymptotes. 

115.  Rectilinear  asymptotes 140 

110.  Asymptotes  parallel  to  the  axes.     Examples 147 

117.  Asymptotes  oblique  to  the  axes.     Examples 148 

Curve  Tracing. 

118.  Examples 151 

Polar  Curves. 

119.  Subtangent  and  subnormal 150 

120.  Lengths  of  the  subtangent  and  tangent 150 

121.  Lengths  of  the  subnonnal  and  nonual.     Examples 157 

122.  Curvature.     Examples 158 

123.  Radius  of  curvature.    Examples 160 

124.  Asymptotes.     Examples 161 

125.  Tracing  of  polar  curves.    Examples 163 


CONTENTS. 


Part  IT.  —  Thk  Integral  Calculus. 


CHAPTER   VI.  — TYPE   INTEGRABLE   FORMS. 

ART.  PAOE 

12(5.  Integral  and  integi-atioii I(i7 

127.  Symbol  of  integi-atioii 107 

128.  Constant  of  integi-ation 108 

129.  Integral  of  a  polynomial 108 

130.  Cadx=<i  C<lx 109 

131.  Type  forms 109 

132.  Remarks  on  the  type  forms.     l']xamples 171 

Elementary   Transformations. 

133.  By  the  introduction  of  a  constant  factor.    Examples 174 

134.  By  the  transference  of  a  variable  factor.    Examples 180 

135.  By  expansion.    Examples 181 

130.  By  division.    Examples 1 82 

137.  By  separation  into  partial  fractions  having  a  connnon  denomi- 
nator.   Examples 182 


CHAPTER   VII.  — GENERAL   METHODS   OP   REDUCTION. 
By  Partial  Fractions. 

138.  Rational  fractions 184 

139.  Case  1.     Examples 185 

140.  Case  2.     Examples 187 

141.  Case  3.     Examples 189 

142.  Case  4.    Examples 192 

By  Rationalization. 

143.  Differentials  containing  surds  of  the  form  .t«,   or  (a  +  hx)'K 

Examples 194 

144.  The  forms  Va  +  bx  ±  x-.     Examples 195 

145.  Conditions  of  rationalization  of  binomial  differentials.    Examples  198 


CONTENTS.  XI 


By  Parts. 

ART.  PAGE 

146.  Formula  for  integration  by  parts 201 

147.  Binomial  differentials.    Formulae  of  reduction.     Examples 201 

148.  Logarithmic  differentials  of  the  form  ar™  (log  .r)«  (/t.    Examples..   208 

149.  Exponential  differentials  of  the  form  x"  c"  (/.r.     Examples 208 

150.  Trigonometric  differentials*  of  the  forms  (I.)  sin^xdx,  cos"  .rc?.r; 

/TT   \  (t^  <JX  /TTT\      Sin"TrfT         COS''.rf/.C         /T\7-    \    *„„m      J 

(II.)    ,    ;    (III.)   ,    ;  (IV.)  tan^a-rtr, 

sin''x     cos"  a:  cos"'x        sin"'.e 

coV^xdx;   (V.)  x"sin(rtx)rf.r,  .r^cos  («x)f/x;  (VI.)  6" sin"  j-rfr, 

e"'  cos"  xdx.     Examples 209 

151.  Circular. differentials  of  the  f onus  / (.r)  siu^ ^ a-f/r,  etc.,  in  which 

fix)  is  an  algebraic  function.     Examples 217 

By  Substitution. 

152.  Trigonometric  differentials  of  the    form   sin"  t  cos"  rrf.r.      Ex- 

amples    218 

158.  Miscellaneous  examples  of  integration  by  substitution 219 

By  Series. 

154.  Integration  by  series.     Examples 221 

Miscellaneous  examples 222 

Successive  Integ^ratioii. 

155.  Examples 224 

Tlie  Constant  of  Integration. 

156.  Indefinite  and  definite  integrals 225 

157.  Definite  integi-als 225 


CHAPTER   VIII.  — GEOMETRICAL   APPLICATIONS. 

158.  Determination  of  the  equations  of  curves.     Examples 229 

159.  Rectification  of  plane  curves.     Examples 230 

160.  Quadrature  of  plane  curves.     Examples 234 

161.  Surfaces  and  solids  of  revolution.     Exami)les 237 


Part   I. 
THE   DIFFEEENTIAL   CALCULUS. 


CHAPTER   I. 

INTRODUCTORY    THEOREMS. 

1.  Quantities  of  the  Calculus.  The  quantities  of  the  Cal- 
culus are,  like  those  of  Analytic  Geometry  : 

Variables :  whose  values  change  continuously  within  the 
limits  assigned  by  their  mutual  relations.  Thus,  in  the  equa- 
tion of  the  circle  a^  +  y^  =  R"^,  x  and  y  are  variables  having 
any  and  all  values  between  the  limits  ±  R. 

Arbitrary  constants :  as  R  in  the  above  eqiiation,  which  may 
have  any  arbitrarily  assigned  values,  but  which  do  not  change 
when  the  variables  change. 

Absolute  constants:  which  admit  of  no  change  whatever; 
such  as  R  would  become  if  the  radius  of  the  circle  were  as- 
sumed to  be  5. 

2.  Functions.  As  in  Analytic  Geometry,  also,  any  quantity 
is  said  to  be  a  function  of  another  when  it  depends  upon  the 
latter  for  its  value.  Thus,  a  —  x,  tana;,  (a^  —  x^)^,  are  func- 
tions of  X.  The  variable  u])on  which  the  function  depends  is 
called  the  independent  variable. 

An  equation  between  two  variables  may  be  solved  for  either 
regarded  as  the  function,  the  other  being  the  independent  vari- 
able. Thus,  from  or  -\-y^  =R'  we  have  y  =  VR^  —  .^•^  in  which  y 
is  the  function  and  x  the  independent  variable,  or  x  =  Vi2^  —  y^, 
in  which  x  is  the  function  and  y  the  independent  variable.  The 
distinction  implies  no  difference  in  the  nature  of  the  variables, 
for  each  is  dependent  upon  the  other,  and  serves  only  to  dis- 
tinguish the  variable  whose  values   are   assigned   from   that 

whose  values  are  derived. 

3 


4  THE   DIFFERENTIAL   CALCULUS. 

A  quantity  may  depend  upon  several  variables  for  its  value, 
and  is  then  said  to  be  a  function  of  two  or  more  variables. 
Thus,  a?  +  y^  —R"^,  xzv,  are  functions  of  two  and  three  variables, 
respectively.  If  no  condition  is  imposed  upon  the  function,  the 
variables  are  said  to  be  independent.  If,  however,  we  subject 
the  function  to  some  condition,  as  cc^  +  ?/-  —  i?-  =  0,  x  and  y  are 
said  to  be  dependent,  since  they  can  only  vary  in  such  a  way 
as  to  make  the  function  zero.  Although  dependent  upon,  that 
is,  functions  of,  each  other,  a  value  may  be  assigned  to  one  and 
that  of  the  other  derived  from  the  equation ;  either  one  may 
therefore  be  regarded  as  the  independent  variable  in  the  sense 
exj)lained  above. 

When  the  variables  are  dependent  and  their  mutual  relations 
are  known,  the  function  may  be  expressed  in  terms  of  any  one 
regarded  as  the  independent  variable.  Thus,  the  function  xzv 
represents  the  volume  of  a  parallelopiped  whose  edges  are  a*, 
z,  and  V,  and  the  variables  are  independent.  If,  however,  we 
impose  the  conditions  x  =  mz,  z  =  nv,  that  is,  if  the  ratios  of 
homologous  sides  are  to  remain  constant,  the  variables  become 
dependent,  and  the  function  may  be  expressed  in  terms  of  any 

a^ 
one,  as  — — • 
mrn 

The  conditions  of  the  problem  will  determine  whether  the 
variables  are  dependent  or  independent,  and  in  the  former  case 
the  manner  of  their  dependence. 

3.   Classification  of  functions. 

I.  Functions  are  classified  as  algebraic  and  transcendental. 
Algebraic  functions  are  those  which  involve  only  the  six  fun- 
damental operations  of  Algebra:  addition,  subtraction,  multi- 
plication, division,  involution,  and  evolution,  the  indices  in  the 
latter  cases  being  constant.  All  other  functions  are  transcen- 
dental ;  the  more  common  of  which  are  : 

The  logarithmic  function,  x  =  log  y,  and  its  inverse  form, 
y  =  a',  the  exponential  function ; 

The   trigonometric  functions,  y  =  sin  x,  y  =  cos  x,  etc.,  and 


LNTKODUCTORY   THEOREMS.  5 

their  inverse  forms,  a;  =  sin"'y,  x  =  cos~^y,  etc.,  the  circular 
functions. 

An  algebraic  function  of  a  single  variable  which  contains  no 
power  of  the  variable  above  the  first  is  called  a  linear  function. 
Such  can  always  be  reduced  to  the  form  mx  +  b. 

II.  If  an  equation  between  several  variables  be  solved  for 
any  one,  the  latter  is  said  to  be  an  explicit  function  of  the 
others,  the  manner  of  its  dependence  being  exhibited  by  the 
solution  of  the  equation.  Otherwise  it  is  said  to  be  an  implicit 
function.  Thus,  in  v?  +  if  =  R^,  x  and  y  are  implicit  functions 
of  each  other ;  while,  in  ?/  =  Vi2^  —  a^,  ?/  is  an  explicit  function 
of  X.  The  difference  is  one  of  form  only,  the  chief  object  of 
Algebra  being  the  reduction  of  functions  from  implicit  to 
explicit  forms.  The  notation  y=f(x),  y=f'(x),  y=(f>(x), 
etc.,  read  'y  a  function  of  a.','  is  used  to  denote  that  y  is  an  ex- 
plicit function  of  x;  and  the  notation  f{x,  y)  =  0,  <f)  (x,  y)  =  0, 
etc.,  to  denote  that  x  and  y  are  implicit  functions  of  each  other. 

III.  If  in  any  function  y=f(x),  y  increases  and  decreases 
with  X,  y  is  called  an  increasing  function  of  x ;  but  if  y  de- 
creases when  X  increases,  or  increases  when  x  decreases,  y  is 
said  to  be  a  decreasing  function  of  x. 

The  increase  and  decrease  referred  to  is  algebraic. 

Thus,  in  y  —  mx  +  6,  y  is  an  increasing  function  of  x ;  but 
in  y  =  —  mx  +  &,  2/  is  a  decreasing  function  of  x.  Again,  in 
y"^  =  2px,  y  has  two  values,  one  of  which 
is  an  increasing,  the  other  a  decreasing,  \ 
function  of  x.  \ 

If  we  plot  the  locus  of  y  z=f(x),  this  "V 

relation  of  the  variables  to  each  other    —     ^■' 


O 


is  represented  graphically.    Thus,  x-  =  y  Fig.  i. 

is  a  parabola  situated  as  in  the  figure, 

from  which  we  see  that  when  x  is  negative,  that  is  in  the  sec- 
ond angle,  aj  is  a  decreasing  function  of  y ;  and  that  when  x  is 
positive,  that  is  in  the  first  angle,  x  is  an  increasing  function 
oiy. 


6  THE   DIFFEKENTIAL   CALCULUS. 

Determine  whether  y  is  an  increasing  or  a  decreasing  function 
ot.in  ,,=  si„.;  ,  =  ta„.;  y^\;  y^a';  y  =  V^?^. 

4.  Increments.  21ie  amount  of  the  increase  or  decrease  of  a 
variable  in  any  interval  of  time  is  called  its  increment,  or  decre- 
ment. It  is  usual,  however,  to  employ  the  word  increment  to 
denote  both  an  increase  and  a  decrease,  the  increment  receiving 
a  negative  sign  where  the  variable  is  decreasing. 

5.  Uniform  change.  A  variable  is  said  to  change  uniformly 
ivhere  its  increment  is  n  umerically  the  same  in  all  equal  intervals 
of  time. 

Since  the  increment  is  numerically  the  same  for  all  equal 
intervals,  the  increment  in  any  interval,  assumed  as  a  unit  of 
time,  may  be  taken  as  the  measure  of  the  change.  This  meas- 
ure is  called  the  rate  of  change,  or  simply  the  rate,  of  the  vari- 
able, and  is  evidently  constant.  Hence  the  rate  of  a  uniformly 
changing  vaHable  is  its  increment  in  a  unit  of  time. 

Representing  by  x  the  total  change  of  the  variable  in  the 
time  t,  and  by  r  the  change  in  the  unit  of  time,  x  =  rt  and 

.■=f;  (1) 

or,  the  rate  of  a  uniformly  changing  variable  is  found  by  dividing 
the  total  change  in  any  time  t  by  t. 

6.  Uniform  motion.  When  the  variable  is  the  distance 
passed  over  by  a  moving  point,  estimated  from  any  origin  in 
the  path,  if  this  distance  changes  uniformly,  the  point  is  said 
to  have  uniform  motion,  and  the  increment  of  the  distance  in 
a  unit  of  time  is  called  the  velocity  of  the  point.  Thus,  if  a 
point  is  said  to  have  a  velocity  of  5  miles  an  hour,  we  mean 
that  its  distance  from  any  point  in  its  path  increases  or  de- 
creases 5  miles  every  hour.  Hence  the  velocity  of  a  point 
having  uniform  m,otion  is  the  rate  of  change  of  the  distance  it 
passes  over.   Representing  the  distance  passed  over  in  the  time 


INTHODUCTORY   THEOREMS.  7 

t  by  5,  and  by  v  the  distance  passed  over*  in  a  unit  of  time, 

-y  =  -)  in  which  v  is  the  rate  of  s. 
t 

7.  Varied  change.  Wlien  the  law  of  change  of  a  variable  is 
such  that  in  no  two  consecutive  equal  intervals  of  time  its  incre- 
ments are  equal,  its  change  is  said  to  be  varied;  and  the  rate  of 
such  a  variable  at  any  instant  is  what  its  increment  would  be  in 
a  unit  of  time  were  the  change  at  that  instant  to  become  uni- 
form. Thus,  if  a  point  so  moves  that  the  increments  of  the 
distance  passed  over  in  consecutive  equal  intervals  of  time  are 
unequal,  its  motion  is  said  to  be  varied,  and  its  velocity  at  any 
instant,  that  is,  the  rate  of  change  of  the  distance,  is  the  dis- 
tance it  would  pass  over  in  a  unit  of  time  were  the  motion  to 
become  uniform  at  that  instant. 

These  definitions  rest  upon  fanailiar  conceptions.  Suppose,  for  exam- 
ple, a  cistern  is  being  filled  with  water  by  a  supply  pipe  in  such  a  manner 
that  the  amount  of  water  supplied  is  the  same  in  all  equal  intervals  of 
time,  this  amount  being  5  gals,  for  one  second.  The  quantity  of  water  in 
the  cistern  (a;)  is  a  variable,  and  the  amount  of  water  actually  supplied 
during  any  interval  is  its  increment ;  and  because  the  change  in  x  is  uni- 
form, we  know  not  only  the  amount  supplied  in  one  second,  but  also  in  any 
other  interval  of  time.  For  unequal  intervals  the  corresponding  incre- 
ments are  unequal,  but  the  rate  at  which  the  Cistern  is  being  filled  is  the 
same  throughout  both  intervals.  The  characteristic  of  uniform  change  is, 
therefore,  a  constant  rate  ;  and  we  say  the  cistern  is  being  filled  at  the  rate 
of  5  gals,  a  second,  or  300  gals,  a  minute,  according  as  the  second  or  the 
minute  is  the  unit  of  time.  If,  now,  the  flow  of  water  through  the  supply 
pipe  ceases  to  remain  uniform,  the  rate  at  which  the  cistern  is  being  filled 
changes,  the  characteristic  of  varied  change  being  a  variable  rate.  In 
both  cases  the  rate  of  the  change  of  the  quantity  of  water  in  the  cistern  is 
an  instantaneous  property  of  that  quantity,  but  in  neither  case  can  we 
measure  it  instantaneously.  When  the  flow  is  uniform,  we  observe  what 
the  actual  change  is  for  any  definite  interval ;  when  the  flow  varies,  we 
ascertain  what  the  change  would  be  for  any  definite  interval  were  the  flow 
to  become  imiform  at  the  instant  considered.  What  these  intervals  are  is 
immaterial ;  but  for  the  comparison  of  rates  it  is  evidently  necessary  to 
adopt  the  same  interval. 

The  following  illustration  is  due  to  Clifford  (Elements  of  Dynamic). 
Suppose  a  train  to  be  moving  from  ^  to  JS  on  a  straight  track,  its  velocity 


8  THE   DIFFERENTIAL   CALCULUS. 

being  the  same  throughout  the  entire  distance.  Then  its  distance  from  A 
is  a  variable,  and  the  distance  passed  over  in  any  interval  of  time  is  the 
increment  of  the  variable.  If  this  increment  is  20  miles  for  one  hour,  we 
know  the  increment  for  one  minute  will  be  i  of  a  mile,  and  that  while 
these  increments  differ,  the  rate  of  change  of  z,  the  distance  from  A,  is  the 
same  during  the  entire  journey.  If  we  now  suppose  a  second  train  is 
moving  in  the  same  direction  on  a  parallel  track,  and  that  it  starts  from  A 
with  a  velocity  less  than  20  miles  an  hour,  but  gi-adually  increasing  to  40 
miles  an  hour ;  and  if  we  suppose  further  that  its  length  is  such  that  some 
part  of  it  is  always  opposite  to  a  traveller  seated  in  the  first  train,  then  it 
will  appear  to  him  to  be  losing  distance  so  long  as  its  velocity  is  less  than 
20  miles  an  hour  ;  but  when  its  velocity  exceeds  20  miles  an  hour,  it  will 
appear  to  be  gaining.  There  must  then  be  some  instant  between  these 
two  states  of  things  at  which  the  second  train  appears  to  the  traveller  to 
be  neither  gaining  nor  losing.  At  that  instant  the  velocity  of  both  trains 
is  the  same,  i.e.  20  miles  an  hour,  or  the  distance  which  the  second  train 
would  pass  over  in  one  hour  were  its  velocity  at  that  instant  to  remain  the 
same  for  one  hour.  In  both  cases,  therefore,  the  velocity  is  determined 
in  essentially  the  same  manner ;  we  suppose  each  train  to  maintain  the 
velocity  it  has  at  any  given  instant  for  a  unit  of  time  and  observe  how  far 
it  goes.     This  increment  is  the  rate  at  that  instant. 

8.  Differentials.  What  ivould  be  the  increment  of  a  variable 
in  any  interval  of  time  loere  its  rate  to  remain  throughout  the 
interval  what  it  teas  at  its  beginning  is  called  the  differential  of 
the  variable.  It  follows  from  Art.  4  that  the  differential  of  a 
decreasing  variable  is  negative. 

The  symbol  for  the  differential  of  any  variable  x  is  dx,  read 
*  the  differential  oi  x.'  The  letter  d  must  not  be  mistaken  for 
a  factor.  Its  meaning  is  '  the  differential  of,'  as  in  sin  x  the 
abbreviation  si7i  means  'the  sine  of.' 

Since  time  (t)  changes  uniformly,  any  interval  of  time  may 
be  represented  by  dt. 

9.  Remark.  The  distinctions  between  the  increment,  differ- 
ential, and  rate,  of  a  variable,  should  be  carefully  observed.  Its 
increment  is  the  amount  of  its  actual  increase,  or  decrease,  in 
any  interval  of  time ;  its  differential  is  wli^t  the  amount  of  its 
increase,  or  decrease,  would  be  in  any  interval  were  its  rate  to 


INTRODUCTORY   THEOREMS.  9 

remain  throughout  the  interval  what  it  was  at  its  beginning. 
Hence  the  increment  and  differential  of  a  variable  are  the  same 
only  when  the  variable  is  changing  uniformly.  Finally,  its  rate 
is  what  the  amount  of  its  increase,  or  decrease,  would  be  in  a 
unit  of  time  were  the  change  of  the  variable  at  any  of  its  values 
to  become  uniform;  a  rate  is  thus  a  particular  differential, 
namely,  the  differential  for  the  unit  of  time. 

10.  Corresponding  increments,  or  differentials,  of  variables 
are  those  which  occur,  or  would  occur,  in  the  same  interval. 

Simultaneous  rates  of  variables  are  their  rates  at  the  same 
instant. 

The  simultaneous  rates  of  variables  which  are  always  equal  are 
evidently  equal. 

11.  Symbol  of  a  rate.     By  Art.  7  the  rate  of  any  variable  x 

at  any  instant,  that  is,  at  any  of  its  values,  is  measured  by  the 

increment  it  would  receive  in  a  unit  of  time  were  its  change  at 

that  instant  to  become  uniform;  hence  if  dx  represents  what 

this  change  would  be  in  any  interval  dt,  we  have  from  Eq.  (1), 

Art.  5,  , 

._  "^ 

'  ~di 

ivhatever  the  interval  dt;  or  the  rate  of  any  variable  is  the  differ- 
ential of  the  variable  divided  by  the  differential  of  t. 

Cor.  Since  the  differential  is  positive  or  negative  as  the 
variable  is  increasing  or  decreasing,  the  rate  of  an  increasing 
variable  is  positive,  and  of  a  decreasing  variable  is  negative. 

Remark.  It  must  be  carefully  noted  that  while  dt  is  arbi- 
trary, for  the  purpose  of  comparing  the  rates  of  different  vari- 
ables, or  of  the  same  variable  at  different  instants,  we  must 
assume  the  same  interval ;  hence  dt  is  constant. 

12.  Corresponding  differentials  of  equals  are  equal. 

Let  y—f(x,  z,  v,  etc.).  Since,  when  two  quantities  are 
always  equal,  their  simultaneous  rates  are  equal  (Art.  10),  if 


10  THE  DIFFERENTIAL  CALCULUS. 

dy  and  d{_f{x,  z,  v,  etc.)]  be  corresponding  differentials  of  y 
iu\df(x,  z,  V,  etc.),  then 

dy  ^  (Z[/(a;,  2,  ^;,  etc.)] 
dt  dt  ' 

or,  since  dt  is  the  same  in  both  members,  dy  =  d[^f{x,  z,  v, 
etc.)].  Hence,  if  an  equation  be  true  for  all  values  of  the 
variables  involved,  the  corresponding  differentials  of  the  two 
members  are  equal. 

^y  =  ^  [/(^)  2>  "^'j  etc.)]  is  called  the  first  derived,  or  first 
differential,  equation  of  y  =f{x,  z,  v,  etc.). 

The  above  is  an  immediate  consequence  of  the  definitions.  For  if  a 
and  3  be  any  functions  whatever,  and  a—  &  for  all  values  of  the  variables 
involved,  the  rates  of  a  and  fi  must  be  the  same  at  any  instant.  Now 
these  rates  are  what  the  changes  in  a  and  0  \vould  be  in  a  unit  of  time 
were  the  common  rate  to  become  constant  at  any  instant ;  and  if  the  rate 
remained  constant  for  any  interval  greater  or  less  than  the  unit,  the  cor- 
responding changes  would  still  be  equal ;  but  these  changes  are  the  differ- 
entials. 

13.  The  immediate  object  of  the  Differeyitial  Calcidus  is  the 
determination  and  comparison  of  the  rates  of  variables. 

The  following  problem  will  serve  as  an  illustration. 

Suppose  a  wheel  to  revolve  about  a  fixed  axis  through  its 

centre,  P  being  any  point  in  the  rim,  and 

that  we  desire  to  compare  the  rate  of  P's  ^^-—r^' 

motion  in  the  arc  AB  with  that  of  its       /  /    \  ^ 

/  /        V '9'  2 

motion  vertically  upward  at  any  instant.     /  /        \ 

This  is  equivalent  to  asking  what  are  the  c    d    a 

rates  of  change  of  the  arc  AP  and  its  sine 

PD.     Hence  if  AP=x,  DP=y,  the '  fundamental  relation  is 

y  =  sin  a*.  Now  if,  as  will  be  shown,  -^  =  cos  x  — ?  the  rate  of 
^  '  '  dt  dt 

y  is  seen  to  be  cos  x  times  the  rate  of  x ;  that  is,  at  any  instant 
the  sine  is  changing  cos  x  times  as  fast  as  the  arc.  If  P  is 
moving  in  the  arc  at  the  rate  of  10  ft.  per  sec,  then  at  A, 
where  cos  x  =  l,  it  is  also  moving  upward  at  the  same  rate. 


INTRODUCTORY  THEOREMS.  11 

At  P,  where  AP=SiVG  of  60°  and  cos  cc  =  cos  60°  =  ^,  it  is 
moving  upward  at  the  rate  of  5  ft.  per  sec,  or  half  as  fast  as 
it  moves  in  the  arc.  At  B,  where  cos  x  =  cos  90°  =  0,  it  is  not 
moving  iipward  at  all. 

14.  Differentiation.  In  the  above  illustration  the  rate  of  y 
is  the  rate  of  sin  x ;  and,  in  general,  the  determination  of  the 
rates  of  variables  involves  the  determination  of  the  rates  of 
the  functions  on  which  they  depend  or  in  which  they  enter. 
Since  the  rate  of  a  variable  is  the  differential  of  the  variable 
divided  by  the  differential  of  t,  the  relation  between  the  rates 
of  variables  will  be  known  when  the  relation  between  their  differ- 
entials is  known. 

The  process  of  determining  the  differential  of  a  function  is 
called  differentiation. 

We  now  proceed  to  determine  rules  for  the  differentiation  of 
the  several  algebraic  and  transcendental  functions. 


CHAPTER   II. 

DIFFERENTIATION  OF  EXPLICIT  FUNCTIONS. 
THE  ALGEBRAIC   FUNCTIONS. 

15.  The  differential  of  a  constant  is  zero. 

This  is  evident  since  a  constant  admits  of  no  change,  and 
therefore  has  no  increment,  whatever  the  interval.  Properly- 
speaking,  such  expressions  as  '  the  differential  of,'  or  '  rate  of  a 
constant '  involve  a  contradiction  of  terms.  But  for  uniformity 
of  expression  it  is  usual  to  say  that  both  are  zero. 

16.  The  differential  of  a  polynomial  is  the  algebraic  sum  of  the 
differentials  of  its  several  terms. 

Let  y  =  X  -i-  z  —  V.  If  the  changes  of  x,  z,  and  v,  at  any  in- 
stant, that  is,  at  any  of  their  simultaneous  values,  become  uni- 
form, the  change  of  y  at  that  instant  will  also  become  uniform ; 
and  therefore,  if  dx,  dz,  dv,  dy,  be  corresponding  differentials  of 
the  variables  and  the  function,  dy=dx-\-dz—dv  (Art.  12).  The 
above  is  evidently  true  of  a  polynomial  of  any  number  of  terms. 

Cor.  ^  =  ^  _,_  ^  _  ^,  or  the  rate  of  the  sum  of  any  num- 
dt      dt      dt     dt 
ber  of  variables  is  the  sum  of  the  rates  of  the  variables. 

Since  the  relation  between  the  rates  is  always  the  same  as 
that  between  the  differentials,  it  will  not  be  necessary  to  repeat 
this  inference  in  the  cases  which  follow. 

17.  The  differential  of  the  product  of  a  variable  and  a  constant 
factor  is  the  differential  of  the  variable  multiplied  by  the  constant 
factor. 

12 


^Q 


Fig.  3. 


B  P 


THE   ALGEBRAIC    FUNCTIONS.    .  13 

Let  y=x-^z-\-v-\-etc.  From  Art.  16,  dy=dx-\-dz-{-dv+ etc. 
Hence  if  x=z=v=etc.,  and  m  be  the  number  of  terms,  y=mx 
and  dy  =  dx  +  dx  -\-  etc.  =  mdx. 

18.  The  differential  of  the  product  of  tivo  variables  is  the  sum 
of  the  products  of  each  into  the  differential  of  the  other. 

Let  y  =  xz.     Then  y  is  the  area  of  a  rectangle  whose  sides 
are  x  and  z.     Let  a,  b,  be  any  two  simultaneous  values  of  x 
and  z  ;  then  at  the  instant  when  x=a 
=  AB  and  2  =  6  =  AD,  we  have  y  =     ^1  iS 

area  ABCD.  Let  BP  represent  what 
would  be  the  change  in  x  in  the  inter- 
val dt  if  at  this  instant  its  change  were 
to  become  uniform,  and  DR  the  corre- 
sponding change  in  z  were  its  change 
also  to  become  uniform  at  the  same  instant.  Then  BP  =  dx, 
DR  =  dz.  The  change  of  y  would  then  also  become  uniform,  and 
for  the  interval  dt  would  be  dy  =  BPQC -{-  DRSC  =  bdx  +  adz. 
But  a  and  6  are  any  simultaneous  values  of  x  and  z.  Hence, 
in  general,  at  any  instant,  dy  =  zdx  +  xdz. 

19.  The  differential  of  the  product  of  any  number  of  variables 
is  the  sum  of  the  products  of  the  differential  of  each  variable  into 
all  the  others. 

Let  y  =  xzv,  and  o:z  =  u.  Then  y  =  uv.  But  dy  =  vdu  -f-  udv 
(Art.  18),  and  du  —  zdx  +  xdz.  Substituting  in  the  former  the 
value  of  du  from  the  latter,  and  of  w  =  xz,  we  have  dy  =  zvdx  -f- 
xvdz  +  xzdv. 

In  the  same  manner  the  theorem  may  be  proved  for  the 
product  of  any  number  of  variables. 

20.  Tlie  differential  of  a  fraction  is  the  denominator  into 
the  differential  of  the  numerator,  minus  the  numerator  into  the 
differential  of  the  denominator,  divided  by  the  square  of  the 
denominator. 


14  THE   DIFFERENTIAL   CALCULtlS. 

Let  y='  •      Then  x  =  yz.      But  dx  =  zdy -\- ydz  (Art.  18). 

_^            ,        dx        dz      dx      xdz      zdx  —  xdz 
Hence  dit  = y—  = s-  =  — —r, 

Z         "^  Z  Z  Z'  z- 

CoR.  1.    If  x  =  a,  a   constant,  then  dx  =  0  (Art.  15),  and 

dy  = 2"  I  ^^'  ^'^^  differential  of  a  fraction  loliose  numerator  is 

constant  is  minus  the  numerator  into  the  differential  of  the  de- 
nominator, divided  by  the  square  of  the  denominator. 

dec 
Cor.  2.    If  z  =  a,  a  constant,  then  dz  =  0^  and  dy  =  — ,  as  it 

1  " 

should  be,  since  y  is  then  -x  (Art.  17). 

21.  The  differential  of  a  variable  having  a  constant  exponent 
is  the  product  of  the  constant  exponent,  the  variable  ivith  its  ex- 
ponent diminished  by  one,  and  the  differential  of  the  variable. 

I.  Let  the  exponent  be  positive  and  integral. 

Then  y  z=x"'  =  xxx  •  •  •  to  n  factors. 

Hence  (Art.  19), 

dy  =  x"~^dx  +  x'^'^dx  +  •••  to  ?i  terms, 
or  dy  =  nx"~^  dx. 

II.  Let  the  exponent  be  a  positive  fraction. 

Then  y  =  a;",  whence  i/"  =  a^"  The  differentials  of  the  two 
numbers  of  this  equation  being  equal  (Art.  12),  we  have,  by  I., 

n?/""^  dy  ■=  maf~^  dx, 

,  ,       m  cc""^  ,        m  ^-1 

whence  dv  = ;  do;  =  —  x"     dx. 

^       n  2/"~  w 

III.  Let  the  exponent  be  negative. 

Then  y  —  x'"  =  —,  n  being  fractional  or  integral.     Clearing 
a;" 
of  fractions,  yx"  =  1 ;  whence,  differentiating  the  product  and 
remembering  that  the  differential  of  a  constant  is  zero. 


THE  ALGEBRAIC   FUNCTIONS.      *  15 

Of"  cly  +  ynx"~^  clx  =  0, 

or  ay  = — - —  =  —  nx~"'^ax. 

^  x" 

-  dx 

Cor.  1.    If  ?i  =  ^,  ?/  =  Vi»,  and  dy  — — -=,  or  tlie  differential  of 

2wx 

the  square  root  of  a  variable  is  the  differential  of  the  variable 
divided  by  twice  the  square  root  of  the  variable. 

Special  rules  might  be  framed  for  n  =  i,  7i  =  \,  etc.,  but  in 
such  cases  the  general  rule  is  preferable. 

Remark.  The  above  method  of  proof  depends  upon  the 
resolution  of  the  power  into  equal  factors,  and  is  therefore 
inapplicable  to  the  case  of  a  variable  having  an  imaginary,  or 
an  incommensurable,  exponent.  The  rule,  however,  as  will 
subsequently  appear,  holds  good  for  these  cases  also. 

Examples.     Differentiate : 

1.  ?/  =  a^  -f  8  a;  —  4  ar". 

dy  =  d(a^  +  3  a;  -  4ar")  =  cZ(ar')  +  d(3'x)  -  (Z(4ar'')    [Art.  IG 
=  2xdx  +  3 da;  -  12 x^dx  =  {2x-\-^ -Viy?) dx. 

[Arts.  21,  15 

2.  y  =  a  +  mx'^  —  lnx^.       dy  =  (m-af~^  —  14:7ix)dx. 

3.  y-  =  2px. 

Although  an  implicit  function,  it  may  be  differentiated 

directly  without  first  reducing  to  an  explicit  form.    Thus, 

P 
d(y^)  =  d(2pa;),  whence  2ydy  =  2pdx,  or  dy  =  —dx. 

4.  a-y-  ±  b-x^  =  ±  a^b-.         dy  =  ^  —  dx. 

d-y 

5.  2/  =  (l4-a^)(l-2ar5). 

dy  =  (1  -  2ar')cZ  (1  +  a;^)  +  (1  +  ^)d  (1  -  2ar=) 

=  (1  -  2ar^)  2a;da;  -  (1  +  ar)  Ga^^da; 

=  2(a;  — 3a;^— ox*)da;;   or  we  may  first  expand   and 
then  differentiate. 


'16  THE  DIFFERENTIAL   CALCULUS. 

6.     y  =  {a  +  b3iP)K 

dy  =  d[ (a  +  ha?) ^]  =  i(a  +  hxY^-dia  +  ha?)  [Art.  21 

= ;  or,  more  expeditiously  by  the  special 

rule  for  the  square  root  of  a  variable  (Art.  20). 
1.  y  =  {ax)  ^  +  hx^.  dy  =  f~  yj^  +  ^  -|)  ^•'^'• 

8.  y  =  {l-\-  x)  Vl  —  X.        dy  =  — ^— "^ —  dx. 

2VI  —  a; 

9.  y  =  Vl  +  V^.  dy  = = —  • 

4Vx  Vl  +  Vx 

Put  the  expression  in  the  form  x^z'^v.     Then 

,       nvx'^''^dx     2x"vdz  ,  afdv 


11.  y  =  Va.-2;-t'J. 

7   _  d^xzH"^)  _  z^v-dx  -\-  2xv'^zdz  +  ^ccz-t;  ■'dt; 

2Va-22'y*  2aj^2v^ 

2^4 do;  ,     1   /-  ,    ,  z-\/xdv 

= ^  4-  i'^  yxdz  -\ — -  • 

2V«  4i;* 

HO                      1  7                        dU; 

12.  y  =  -  dy  =  -~- 

X  xf- 

HO             a  7            "dx 

13.2/  =  —:.  ^^y  =  -r^- 

Voj  2a;:i 

H  ,            VaJ  ,          do; 

14.  ?/  =  -^  •  dy  = 


2  '       4V 


a; 


H  ^  1  +  a;  7  2  dx- 

lo.  y  =  z~ —  <^^y  =  -r, r»- 

^      1-x  0--^) 

(1  -  a;)d(l  +  a;)  -  (1  +  a;)d  (1  -  a;) 
dy  = ^^^y 

^      a-x  ^      {a-xf 


THE  ALGEBRAIC  FUNCTIONS. 


(1  +  x)" 

f  ^  Y 

Vl  -  x) 

2abx' 

(1  +  ax^y^ 

2  +  mx  +  x^ 

dy  = 

{l  +  xy+' 

■dx. 

dy  = 

2xdx 

a-xy 

d>f  = 

2abx\A- 

a3?)dx 

17.  y=       ''■' 


18.  y  = 

19.  «  = 

(1  +  aa^)2 

on  2  +  WIX  +  X^  -,  c(  l\j 

20.  y  =  — ! ! dy  =  21  X ]dx. 

x  \       x^J 

2 
Put  the  expression  in  the  form  y  =  -  +  m  -\-x-. 

21.  y=     ,  '  dy  = 


Put  the  expression  in  the  form   2/  =  2(l  — ar)~^   and 
differentiate  as  a  power  rather  than  as  a  fraction. 


oo           Vl  +  ar  ,  dx 

22.  y  = ^ —  dy  =  - 


«  X- Vl  +  ^ 

23.  y  = dy=    ^^^^ 


{1-^y  (1-a^)' 


24.  y=J^.  dy  =  -  '^"^ 


\T 


+  ^*  (l  +  a;)Vr^^ 


o-  L    ,  a  ,   6  ,  1         ax  +  2b 

2o.  y  =  ^1  +  -  +  -.  dy  =  -  — -  ^ 

\        X      or  2ar -w/'i.2i  „^_i 


..^  da;. 

X      a^  2x^  Va;2  +  ax  +  6 


r>p       _    Va  +x  ,   _        Va(  Vx  —  Va)c?x 

Va  +  Vx  2  V.x  Va  +  aj  ( Va  +  Vx)^ 


27.  y  =  ^^ dy/  =  I  ^'^"^^•^'  +  3a^  I  dx, 

Vl  +  03^  —  X  <-  Vl  +  ar^  ^ 


-ar 
nationalize  the  denominator  before  differentiating. 


28.  v^Vl-'^-^.  eZ,  =  2J     ^"^l-^-^^lLUa^. 

Vl-x2^^  C  (l-2x2)Vl-x'^ 

29.  1/  =  ^^.  dy  =  -  ^-t^  dx. 

Vx  2x2 


18  THE   DIFFERENTIAL   CALCULUS. 

30.  v=_l_  +  __i dy  =  l( ^ ^ \dx. 

Vl+x      ^/l-x  2V(i_a;)'      (l+a;)V 

31.  y  = ^=-  dy=(l-\-—^:^)dx. 

X  —  -Vaf  —  c  \        Va^  —  cj 

32.  y  =  - -^^^ dy  =  -a^^(^  +  ^')  +  ^^^^-^^)da;. 

OQ    „,  1  ^  if  —  Vl  —  a^  7 

d3.  ?/  = •  dy  =  — ^ — ^z=:-  dx. 

x+^l-a?  Vl-x2(l  +  2a;Vl-a;') 

35.  y  =(2a*+a;^)(a^+a;^)i  d^  =      ^a- +  Ja;-_^^^ 

4:X^- {a^-  +  x^Y 

36.  y  =  {x-  VT^^^)".    dy  =  w  («  -  Vl^^) "  ^  ^^-^'+^'  (^3.. 

Vl— a^ 


37.  y  =  — ^zi=r^ .  dy  = ,    1  H ==    dec. 

Vl  +  a'-Vl-a^  ar'V        Vl-W 

22.  Analytic  siffnification  of  the  ratio  —  • 

rfa? 

Let  2/=/  (a;).  The  only  variable  which  enters  the  function 
being  x,  the  function  y  will  change  only  as  its  variable  x 
changes,  and  the  rate  of  change  of  y  will  depend  upon  the 
rate  of  change  of  x.  Let  k  be  the  ratio  of  these  rates  at  any 
instant.     Then 

dy      ^  dx      .  dy 

-Tr  =  K-rr,  whence  -V-  =  a;. 

dt         dt  dx 

Hence  the  ratio  of  the  differential  of  the  function  to  the  dif- 
ferential of  the  variable  is  the  ratio  of  the  rate  of  change  of  the 
function  to  that  of  the  variable.  It  is  evidently  independent  of 
the  interval  dt. 


THE   ALGEBRAIC   FUNCTIONS.  19 

As  derived  by  differentiation  from  the  function  y  =f  (x), 
this  ratio  is  called  its  first  derived  function,  or  simply  its  first 
derivative ;  also,  being  the  factor  k  by  which  the  rate  of  the 
variable  is  multiplied  to  obtain  the  rate  of  the  function,  it  is 
called  the  first  differential  coefficient. 

CoR.  If  ^  is  positive,  y  is  an  increasing  function  of  x; 
and  if  negative,  y  is  a.  decreasing  function  of  x  (Art.  11,  Cor.). 

23.  Applications.  1.  Compare  the  rates  of  change  of  the 
ordinate  and  abscissa  of  the  parabola  whose  parameter  is  4. 
Which  is  changing  most  rapidly  at  the  point  x  =  d?  Where 
are  they  changing  at  the  same  rate  ?  If  at  the  point  x  =  16 
the  abscissa  is  increasing  at  the  rate  of  24  ft.  a  second,  at 
what  rate  is  the  ordinate  then  changing  ? 

2      ,  dy     2      ..  .  .  .  ^      dy     2  dx 

^  =^^'  •••  dx  =  y^  ^^^^^'  "^^^  ^^  '^''^^''''dt=ydt'  °^'  "' 
general,  the  ordinate  changes  -  times  as  fast  as  the  abscissa. 

2^1 

For  x  =  9,  y  =  ±6,  and  -  becomes  ±  o ,  showing  that  the  or- 

y  -^  '^ 

dinate  is  increasing,  or  decreasing,  k  as  fast  as  the  abscissa  at 

the  points  (9,  6),  (9,-6).     When  the  ordinate  and  abscissa 

are  changing  at  the  same  rate,  we  must  have  -^  =  -=1, .:  y  =  2, 

which,  in  the  equation  of  the  curve,  gives  x  =  l.  This  is  as  it 
should  be,  for  (1,  2)  is  the  extremity  of  the  parameter,  at  which 
point  the  generating  point  is  moving  in  the  direction  of  tlie 
focal  tangent,  whose  inclination  to  the  axis  of  X  is  known 

2  1 

to   be   45°.      For    a;  =  16,    ?/  =  ±  8,    and  -  becomes  ±  -j,   or 

dy         Idx    ,  .,  ,  y         .  ^ 

jt  =  ±  J  TT  ;  hence  if  at  x  =  16  the  abscissa  is  increasing  at 

the  rate  of  24  ft.  a  second,  the  ordinate  in  the  first  angle  is 
increasing,  and  in  the  fourth  angle  decreasing,  at  the  rate  of 
^  X  24  =  6  ft.  a  second. 

2.  Compare  the  rates  of  change  of  x  and  y  in  the  ellipse 
ahf  -j-  6V  =  a^6l     Is  y  an  increasing  or  decreasing  function  of 


20  THE   DIFFERENTIAL   CALCULUS. 

ic?     Compare  the  rates  at  the  extremities  of  the  axes.     If  the 

(5 
axes  are  6  and  4,  at  what  point  is  y  changing  2-y/-  as  fast  as  a;? 

-^  = s- ,  and  is  negative  when  x  and  y  have  like  signs ; 

dx         a^y  °     ' 

hence  the  function  is  decreasing  in  the  first  and   third,  and 
increasing  in  the  second  and  fourth,  angles.     At  the  extremi- 

dy 
ties  of  the  transverse  axis,  y  =  0,  and  ~  =  oo,  or  y  is  changing 

infinitely  faster  than   x.      To   determine  the   point  where  y 

\5  b^x         4x  fS 

changes  2\  -  as  fast  as  x,  we  have = =2  v-,  which, 

~3  a-y         *dy         ^3 

with  the  equation  of  the  curve  9i/^  +  4a^  =  36,  gives  four  points 

3    .—  1 

whose  coordinates  are  numerically  j  V lo  and  ?,• 

3.  The  altitude  of  a  right  triangle  increases  at  the  rate  of 
10  in.  a  second.     At  what  rate  is  the  area  increasing  ? 

Let  h  =  base,  x  =  altitude,  y  =  area.    Then  y  =  — ,  .-.  -^  =  -, 

2        dx     2 
which  is  a  constant ;  therefore  the  area  increases  uniformly  at 

the  rate  of  -  X  10  =  5  6  sq.  in.  a  second. 

Li 

4.  A  spherical  balloon  is  being  filled  with  gas  at  the  rate  of 
m  cub.  ft.  a  second.  At  what  rate  is  the  diameter  increasing 
when  its  length  is  6  ft.? 

Let  y  =  diameter,  x  =  volume. 

Then        X  =  ^/,    or    j,  =  f^J;  .-,  f?  =  {l^. 
6  VV  (^'^       \97rar 

When      y  =  Q,   x  =  36  tt,   and   ^  = 

•^        '  dx      IStt 

Hence  when  the  diameter  is  6,  it  is  increasing  at  the  rate  of 
q-5—  ft.  a  second. 

loTT 

5.  A  rectangle  whose  sides  are  parallel  to  the  axes  is  in- 
scribed in  the  ellipse  a^y^  +  b-ay'  —  o?b^.  Compare  the  rates  of 
change  of  its  area  and  side. 


THE   ALGEBRAIC    FUNCTIONS.    ^"-^  21 


Let  X  and  y  be  the  half  sides  and  z  the  area.  Then  2;  =  4  ccy. 
To  eliminate  y,  and  so  obtain  z  a  function  of  a  single  variable, 
we  have  from  the  equation  of  the  ellipse, 

46    ,-^0 i    dz      Ab   a- -2a? 


?/  =  -Va— ar,  .•.z  =  —  -Va-ar  —  x*,    —  = 
a  a  dx 

In  a  similar  manner  we  may  find  the  ratio 


a  a  dx       a    -^Jq^  _  jp2 

dz 
dy' 


6.  The  radius  and  altitude  of  a  right  cone  vary,  the  slant 
height  remaining  constantly  25  ft.  Compare  the  rates  of 
change  of  the  volume  and  altitude.  When  the  altitude  is  4 
ft.  and  changing  3  ft.  a  second,  how  fast  is  the  volume 
changing  ? 

Let  s  =  slant  height,  x  =  altitude,  z  =  radius  of  base,  and 

y  =  volume.     Then  y  = To  eliminate  z  we  have  the  con- 

o 
dition  z^  +  0?  =  ^,  .:  z-  =  s-  —  xr  and  y  =  ^  {^x  —  x^).    Whence 

-f-  =  Z  (^—  3af)=  ^  (625  —  3x^) ;  that  is,  the  volume  is  increas- 

ing  ^  (625  —  So?)  times  as  fast  as  the  altitude  is  in  linear  feet. 

577 

When  a;  =  4,  this  becomes tt,  and  at  that  instant  the  volume 

o 

is  changing  at  the  rate  of  577  tt  cub.  ft.  a  second. 

The  student  will  observe  that  he  may  eliminate  before  or 
after  differentiation.     Thus,  differentiating  first, 

dy  =  '^{2zxdz  +  z'dx),  .■/^  =  l(2zx'^+z'). 
o  dx      3  dx 

But  from  z^-\-a?  =  s%  —  = ;  substituting  this  value  with 

dx  z 

those  of  z  and  z-,  ^-^  =  -  (^  —  3ar),  as  before. 
dx      3 

7.  A  point  P  moves  from  ^  at  a  uniform  rate  in  the  direc- 
tion of  AP,  at  right  angles  to  AB.  A  light  C,  whose  intensity 
at  a  unit's  distance  is  125,  is  vertically  over  B.  If  AB  =  10, 
BC  =  5,  compare  the  rates  of  the  motion  and  illumination  of 


22  THE  DIFFERENTIAL   CALCULUS. 

P  (understanding  that  the  intensity  of  a  light  at  any  point  is 
its  intensity  at  a  unit's  distance  divided  by  the  square  of  the 
distance),  when  AP=  10. 

Let  AP  =  X.     Then  tlie  illumination  at 

„  125  125 


GP-      10-  +  52  +  a;2 

d//^  250  a; 

'■  dx~      (125  +  0^)2' 

which,  when  x  =  10,  becomes  —  ^j,  or  the  rate  of  change  of 
the  illumination  of  P  is  ^  times  the  rate  of  change  of  AP, 
and  is  decreasing. 

8.  If,  in  Fig.  4,  BC  is  a  lamp-post  10  ft.  high,  and  a  man 
whose  height  is  6  ft.  walks  from  B  in  the  direction  BA  at  the 
rate  of  3  miles  an  hour,  show  that  the  extremity  of  his  shadow 
moves  at  the  rate  of  7^  miles  an  hour. 

9.  In  Ex.  8  show  that  the  length  of  the  man's  shadow  is  in- 
creasing at  the  rate  of  A^  miles  an  hour. 

10.  In  Fig.  4  show  that  if  the  man  walks  from  A  towards  B, 

y 

he  is  approaching  B  -  times  as  fast  as  he  is  approaching  C, 
where  BA  =  x,  CA  =  y. 

11.  A  ship  is  sailing  northeast  at  the  rate  of  10  miles  an 
hour.     At  what  rate  is  it  making  north  latitude  ? 

Ans.    5V2  miles  an  hour. 

12.  The  area  of  a  circular  plate  of  metal  is  expanded  by 
heat.  Find  the  rate  of  change  of  the  area  when  the  radius  is 
5  in.  and  increasing  .01  in.  a  sec.  Ans.   -^^ir  sq.  in.  a  sec. 

13.  If  the  thickness  of  the  plate  of  Ex.  12  increases  one- 
half  as  fast  as  the  radius,  find  the  rate  of  increase  of  the  volume 
when  the  radius  is  5  in.  and  the  thickness  .5  in. 


THE   ALGEBRAIC    FUNCTIONS.  23 

Let  V  =  volume,  x  =  radius,  y  =  thickness.  Then  v  =  njc^y ; 
whence 

dv      o         dx  ,       ^  dy      ,o  ,   ,      ,v  dx       7  ,     . 

—  =  ZttX)! h  irxr  -^  =  CzTTxy  +  iTrar)  —  =  —  n  cub.  in.  a  sec. 

dt  dt  dt  dt      40 

14.  Two  ships,  on  courses  whose  included  angle  is  60°,  are 
sailing  away  from  the  intersection  of  the  courses  with  veloci- 
ties of  6  and  4  miles  an  hour.  Find  the  rate  at  which  they 
are  separating  when  10  and  15  miles  respectively  from  the 
intersection. 

Let  z  =  distance  between  the  ships,  x  and  y  their  distances 
from  the  intersection,  6  and  4  being  the  rates  of  x  and  y 
respectively. 

Then        z-  =  x^ -{-y- —  2xy  Go%Q(f  =  3? +  y- —  xy, 


dz 
dt 


^         -^^  dt      ^  -^        ^  dt 


55  21- 

Vl75  "'        X^ 


15.  Find  the  rate  of  separation  in  Ex.  14  under  the  suppo- 
sition that  the  ships  start  together  from  the  intersection  of 
the  courses,  with  the  velocities  6  and  4. 

22  =  (60'  +  (40'-24f-,  .-. -=V28. 

16.  C  is  any  point  without  a  circle  whose  centre  is  0,  and 
OC  cuts  the  circle  at  A.  Find  the  relative  rates  of  departure 
from  C  and  OC  of  a  point  P  moving  from  A  in  the  arc  of  the 
circle. 

Let  P  be  the  position  of  the  point  at  any  instant,  y  =  PM, 
the  perpendicular  on  OC,  PC=x,  OC=a,  OA  =  B.     Then 


pc=v<;.'M'  +  PM% 


or  x=-^(a-VW^yy  +  f, 

xVR'  -  y' 


24  THE   DIFFERENTIAL   CALCULUS. 

24.  Geometric  signification  of  — • 

dx 

Since  to  every  equation  y=f(x)   there  corresponds  some 

plane  locus,  the  ratio  -^  is  evidently  capable  of  geometric 
interpretation. 

Let  M'N'  be  the  locus  of  y  =f(x),  and  P  the  position  of  the 
generating  point  at  any  instant.-  Then  dx,  dy,  being  corre- 
sponding differentials  of  x  and  y,  are 
what  would  be  the  changes  in  x  =  OD 
and  y  =  DP  during  any  interval  if  at 
its  beginning  their  rates  of  change 
should  become  constant.  But  this 
will  evidently  be  the  case  if  at  P 
the  motion  of  the  generating  point 
should    become    uniform    along    the  Fig.  5. 

tangent  at  P.     Hence  PQ,  QB,  being 

what  would  be  the  corresi)Onding  increments  of  x  and  y  in  any 
interval  dt  if  the  change  of  each  became  uniform  at  the  instant 
considered,  are  corresponding  differentials  of  x  and  y,  and 

|^  =  ^  =  tanXrP=tana.  (1) 

PQ      dx  ^  ^ 

The  tangent  of  the  angle  made  by  any  straight  line  with  the 
axis  of  X  is  called  the  slope  of  the  line.  As  the  tangent  at 
any  point  of  a  curve  has  the  direction  of  the  curve  at  that 
point,  the  slope  of  a  curve  is  that  of  its  tangent ;  hence  the 

dy 
value  of  -j-,  ft^  ctwy  instant^  that  is,  for  any  simidtaneous  values 

of  X  and  y,  measures  the  slope  of  the  curve  at  the  corresponding 

point. 

In  the  figure,  y  is  an  increasing  function  of  x,  a  is  an  acute 

dy  .  .   . 

angle,  and  tan  a,  or  -^^  is  positive.     In  the  vicinity  of  M',  how- 

ever,  y  is  a.  decreasing  function  of  x,  a  is  an  obtuse  angle,  and 

tan  a,  or  -p?  is  negative,  as  already  seen  in  Art.  22. 


THE  ALGEBRAIC   FUNCTIONS.  25 

It  is  evident  that  the  slope  will  in  general  vary  from  point 
to  point,  and  the  first  derivative  will  therefore  be  in  general  a 
function  of  x;  but  that  for  any  particular  value  of  x  it  has 
a  definite  value  independent  of  dt,  that  is,  independent  of  dx, 

since  from  the  similar  triangles  PQR,  PQ'E',  -j-  remains  con- 
stant, whatever  the  interval. 

25.  Relations  between  the  velocities  in  the  path  and  along 
the  axes. 

Let  s  =  distance  passed  over  by  the  generating  point,  esti- 
mated from  any  point  in  its  path,  that  is,  the  length  of  the 
path.  Since,  when  the  changes  in  x  and  y  (Fig.  5)  become 
uniform,  the  generating  point  moves  in  the  direction  of  the 
tangent  PR,  PR  =  ds,  PQ  =  dx,  QR  =  dy,  are  corresponding 
differentials  of  s,  x,  and  y ;  and  from  the  right  triangle  PQR, 

ds'-  =  dx^  +  dy\  (1) 

Hence  if  y=f{x)  be  the  path  of  a  moving  point,  —  is  the 

rate  of  change  of  the  distance,  or  the  velocity  of  the  point  in  its 

path;   and,  for  like  reasons,  — >  -^,  are  its  velocities  in  the 

dt    dt 

directions  of  the  axes. 

By  differentiating  y=f(x)  we  can  compare  the  horizontal 

dor         (In 
and  vertical  velocities,  and  substituting  either  —  or  -^  from 
'  ^  dt        dt 

the  differential  equation  of  the  path  in 


dt      \[dtj      [dtj 


we  can  compare  the  velocity  in  the  path  with  either  the  hori- 
zontal or  vertical  velocity. 

dx  dv 

Since  -j-  and  -—  are  distances,  namely,  the  distances  which 
etc  uz 

the  point  would  pass  over  in  a  unit  of  time  in  the  directions  of 

the  axes  if  its  velocity  in  each  direction  became  uniform,  they 


26  THE   DIFFERENTIAL   CALCULUS. 

are  positive  or  negative  according  as  each  is  in  the  positive  or 
negative  direction  of  the  corresponding  axis. 

26.  The  following  relations  will  be  found  of  use  hereafter. 
Let  PN  (Fig.  5)  be  the  normal  at  P.     Then 

cosa=      sin<^  =  —  1  (1) 

ds 

dy 
sm  a  =  —  cos  (^  =  -j--  (2) 

27.  Applications.     1.  To  find  the  general  equation  of  a  tan- 
gent to  any  plane  curve. 

Let  y  =  /(a-')  be  the  equation  of  the  curve  and  {x',  y')  the 
point  of  tangency.     The  equation  of  a  straight  line  through 

dv 
(x',  y')  is  y  —  y'  =  m  (x  —  x').    If  we  form  ~  from  the  equation 

of  the  curve,  and  substitute  in  it  the  coordinates  of  the  given 

point  of  tangency,  we  have  the  slope  of  the  curve  at  this  point 

(Art.  24).     But  the  slope  of  a  curve  at  any  point  is  that  of  its 

cly  dv 

tangent  at  that  point ;  hence,  representing  by  -—,  what  -y-  be- 

dy' 
comes  for  the  point  (x',  y'),  and  substituting  -^,  for  m, 

2.   Deduce   the   equation  of    the    tangent    to    the    ellipse 
ay  +  h'x"  =.a'b^ 

From  the  equation  of*  the  ellipse,  -^  = ^j  the  general 

expression  for  the  slope.     For  the  particular  point  (x',  y')  this 

b'x' 
becomes -— >  and  the  equation  of  the  tangent  is,  therefore, 

dy 

b^x' 

y  —  y'  = -—  (x  —  x').    Clearing  of  fractions  and  svibstituting 

a-y' 

for  a^y'-  +  b'x'^  its  value  a^6^,  the  equation  assumes  the  simpler 

form  a^yy'  -|-  b'xx'  =  aH)^. 


THE   ALGEBRAIC    FUNCTIONS.  27 

Show  that  the  equation  of  the  tangent  to : 

3.  The  hyperbola  aV  —  ^'^  =  —  «"^^  is  a^yy'  —  b-xx'  =  —  a-b^. 

4.  The  parabola  y-  =  22JX  is  yy'  =  p(x  +  x'). 

5.  The  circle  y^  -{-x^=  R-  is  yy'  +  xx'  =  R-. 

6.  The  circle  y^=2  Rx  —  a^  is  y  —  y'  =  — ~ —  (cc  —  x'). 

7.  The  hyperbola    xy  =  in,    referred   to   its   asymptotes,  is 

8.  The  cissoid  f  =      ^      is  y-y'  =  ±  ^•''(3«-^')  (x^x'). 

2«-^  (2a-a;')' 

9.  The  curve  a^y"  -f-  6''ar'  =  orW  is  y  —  y'  = —  (x  —  x'). 

a^y'^ 

10.  Find  the  slope  of  y^  =  2px  at  the  vertex  ;  at  the  extremi- 
ties of  the  parameter.  Is  the  generating  point  ever  moving  in 
a  direction  parallel  to  X  ? 

cly      p 
From  y^  =  2px,  -^  =  -,  which  is  oo  for  y  =  0.     Hence  the 

tangent  is  perpendicular  to  X  at  the  vertex.     For  y  =  ±p, 

-;-=  ±1,  the  slope  of  the  focal  tangents,  which  therefore  make 
ctx  1 

civ 
angles  of  45°  and  135°  with  X     Since  -^  is  zero  only  when 

y  =  <X),  .-.  x  =  cc,  there  is  no  point  at  which  the  tangent  is 
parallel  to  X. 

11.  Find  the  slope  of  y=x'^ -2x^  +  3  at  x=l;  x  =  3; 
x  =  -2.  Ans.  0;  9G ;   -24. 


12.  At  what  point  of  y-  =  ax^  is  the  slope  0  ?  1  ? 

Ans.    (0,0);   ^^ 


9  a  27  a 


28  THE   DIFFERENTIAL   CALCULUS. 

13.  Find  the  equation  of  the  tangent  to  the  parabola  y^=  2px 
inclined  30°  to  X. 

Since  the  angle  is  30°,  its  slope  is  — ^,  and  we  must  have 
„        1  ,-  V3 

--  =  — =?  or  y'  =pv3.    Hence  from  the  equation  of  the  curve, 

y      V3 

x' =  '-^p.     Substituting  these  values  in  yy'  =p(x-\-x'),  we  ob- 

,   .    "         1        ,  V3 
tain  y  =  — -  x  -\ p. 

V3  2 

14.  At  what  angle  does  1/^=12 a;  intersect  y^-\-o(^-\-6x—6S=0? 

p 

The  points  of  intersection  are   (3,  ±  G) ;  the  slopes  are  -, 

3 +x  y 

J  which  become,  for  the  point  (3,  6),  1  and  —  1.    These 

being  negative  reciprocals  of  each  other,  the  curves  intersect 
at  (3,  6)  at  an  angle  90°. 

15.  Show  that  the  cissoid  cuts  its  circle  at  an  angle  whose 
tangent  is  2. 

IG.  Show  that  the  length  of  the  tangent  to  the  hypocycloid 
x^  ^y3  —a^  intercepted  between  the  coordinate  axes  is  con- 
stant. 

The  equation  of  the  tangent  is  —  +  -^  =  a*. 

x'^     y'^ 

17.  To  find  the  general  equation  of  the  normal  to  any  plane 
curve. 

The  normal  passes  through  the  point  of  contact  and  is  per- 
pendicular to  the  tangent.  The  condition  of  perpendicu- 
larity is  m'  = Hence  the  equation  of  the  normal   is 

cly' 

18.  Find  the  equations  of  the  normals  to  the  conic  sections  : 

*?     f  f 

Ellipse,  y-y'=^{x-x');  Parabola,  y -y' =  ~~(x-x')  ; 

t  2    t 

Circle,  2/  =  | a? 5  Hyperbola,  y -y' =  -^,  {x-x'). 


s 
THE    ALGEBRAIC    FUNCTIONS.  29 

19.  Find  the  equations  of  the  taugent  and  normal  to  ^/^  =  9a^ 
at  the  point  (1,  3).  Ans.  y  =  |a;  —  | ;  y  =  —  \x  -\-  ^-. 

20.  To  find  the  lengths  of  the  subtangent  and  tangent  to 
any  plane  curve. 

.  In  Fig.  5, 

t&n  DTP     cly'      ^  dy' 


dx' 


TP=VTD'  +  DP'-  =^,"+(y|^'=,'^[r^, 


21.  To  find  the  lengths  of  the  subnormal  and  normal  to  any 
plane  curve. 

In  Fig.  5,  DN=  PD  tan  nPN=  PD  tan  DTP  =  y' ^. 


^^-■>Ri5"- 


22.  Find  the  subtangents  and  subnormals  of  the  conic  sec- 
tions. ♦ 


SCBTANQBNT. 

Subnormal. 

Ellipse, 

x^  -  a? 
x' 

a? 

Hyperbola, 

a;'2  -  a? 
x> 

b^x' 
a' 

Circle, 

y" 

x'' 

-x'. 

Parabola, 

2x'. 

P' 

The  signs  may  be  neglected  if  lengths  only  are  required. 
The  sign  will,  however,  indicate  the  direction  if  the  subtangent 
and  subnormal  be  reckoned  respectively  from  T  and  D,  Fig.  5. 

23.  Prove  that  the  subtangent  of  the  hyperbola  xy  =  m  is 
the  abscissa  of  the  point  of  contact,  and  that  the  subnormal 
varies  as  the  cube  of  the  ordinate. 

,dx'  ,       ,dy'         y'^ 

^  dy'  dx'         m 


30  THE  DIFFERENTIAL   CALCULUS. 

24.  Prove  that  the  subtangent  of  the  serai-cubical  parabola 
if  =  a'3?  is  two  thirds  the  abscissa  of  the  point  of  contact, 
and  that  the  subnormal  varies  directly  as  the  square  of  the 
abscissa. 

25.  A  point  moves  with  a  constant  velocity  m  in  the  arc 
of  the  parabola  2/^=  8  a;.  Pind  the  velocities  in  the  directions 
of  the  axes  when  cc  =  8. 

From  ?/-  =  8  a;,  we  have  —  =  — - ,  and  by  condition  —  =  m. 
dt      y  dt  ^  dt 

Substituting  these  values  in 


ds 
di 


=m'^($} 


we  obtain 


dx  _       my  ^    dy  _4:  dx  _       Am 

^^       V/-I-16'       "   dt~  y  dt  ~  -yjy-i  _|_  le 


For  X  =%,.-.  y  =  8,  these  become  — -  and  — - ;  hence  at  the 

V5  V5 

point  (8,  8)  the  horizontal  and  vertical  velocities  are  as  2  to  1, 

2  1 

and  are  — ^  and  — r.  times  that  in  the  path. 

V5  V5 

26.  The  orbit  of  a  comet  is  a  parabola,  the  sun  occupying 
the  focus.  Compare  the  velocity  of  the  comet  with  its  rate  of 
approach  to  the  sun. 

The  distance  of  any  point  of  the  parabola  from  the  focus  is 

rz=x+^,  .'.  —  =  — ,  or  its  rate  of  approach  to  the  sun  is  the 

2         dt      dt 
same  as  its  horizontal  velocity.     But,  as  shown  in  Ex.  25, 

—  =  —  ^        —  •     Hence,  in  general,  its  rate  of  approach  to 
dt      V2/'+p'  f^^  5       &  »  i^i- 

the  sun  is  —  ^        times  its  velocity.     At  the  vertex,  y  =  ^ 

and  ■ —  =  —  =  0,  or,  at  the  vertex,  it  is  not  approaching  the 

dr        1     ds 
sun  at  all.     When  y  =p,  —  =  — -  —  •     When  at  a  distance 

dt      -^2  <^i 


THE   TRANSCENDENTAL   FUNCTIONS.  31 

from  the  sun  equal  to  the  parameter  of  the  orbit, 

r  =  2»  =  x  +  ",    .-.  a;  =  -»  and  w=Vo»,  and    —  =-V3  — 
^2  T  ^  ^  dt      2         dt 

27.  A  point  moves  in  the  arc  of  the  circle  a?  -\-y-  =  25,  and 
has  a  velocity  10  in  passing  through  the  point  (3,  4).  Show 
that  its  velocities  in  the  directions  of  the  axes  are  8  and  G, 
numerically. 


THE    TRANSCENDENTAL    FUNCTIONS. 

The  Logarithmic  and  Exponential  Functions. 

28.  The  logarithmic  function. 

Let  x  =  ny,  (1) 

n  being  any  arbitrary  constant.    Then 

log„x  =  log„w  +  log<,?/,  (2) 

in  which  a  is  the  base  of  the  logarithmic  system. 
Differentiating  (1)  and  (2), 

dx  =  ndy, 

d  (log^x)  =  d  (log^y)  ; 

and,  by  division,     iO^^^LO^.  (3) 

'    •'  '  dx  ndy  ^  ' 

Eliminating  n  from  (3)  by  substituting  its  value  from  (1), 

dx  dy      ^ 

X  y 

clcc 
or  the  ratio  of  d  (log^x)  to  —  is  the  same  as  that  of  d  (log^y) 

dy      „.  .  ^ 

to —      Since  n  is  arbitrary,  the  ratios  in  (4)  are  constant. 

Let  m  be  this  constant.     Then 

d(log„x)  =  m^^ 

X 


32  THE   DIFFERENTIAL   CALCULUS. 

Now  the  only  quantities  involved  in  any  logarithmic  system 
are  the  number,  its  logarithm,  and  the  base.  Since  of  these 
the  two  former  are  variable,  while  m  is  constant,  m  must  de- 
pend upon  the  base.  The  value  of  m  corresponding  to  any 
base  is  called  the  modulus  of  that  system.     Hence 

The  differential  of  a  logarithm  of  a  variable  is  the  modulus  of 
the  system  into  the  differential  of  the  variable  divided  by  the  vari- 
able. 

The  relation  between  the  modulus  and  the  base  of  any  sys- 
tem will  be  established  later  ;  but  as  the  only  system  employed 
in  analytic  investigations  is  that  whose  modulus  is  unity,  called 
the  Naperian  system,  the  above  rule  becomes  : 

The  differential  of  the  Napenan  logarithm  of  a  variable  is  the 
differential  of  the  variable  divided  by  the  variable. 

Unless  otherwise  mentioned,  by  log  x  will  hereafter  be 
meant  Naperian  logarithm  of  x.  The  base  of  this  system  is 
represented  by  the  letter  e,  and  its  value  will  be  shown  to  be 
2.718281. 

29.  The  exponential  function. 

I.    When  the  base  is  constant. 

Let  y  =  a".     Then,  in  the  system  whose  base  is  b, 

log^y  =  xlog,,a. 
Differentiating  both  members, 

•m—  =log^adx, 

y 

.                                     ,        a''  logft  adx 
whence  a?/  = >  -- 

^  m 

or   the   differential  of  an   exponential  function  whose   base  is 
constant  is  the  function  into  the  logarithm  of  the  base  into  the 
differential  of  the  exponent,  divided  by  the  modulus  of  the  system. 
For  the  Naperian  system,  m  =  17  and  we  have 

dy  =  a"  log  adx. 


THE   TRANSCENDENTAL   FUNCTIONS.  33 

If  the  exponential  base  is  also  the  base  of  the  logarithmic 
system,  log  a  =  1,  and 

dy  =  a*  dx, 

or,  e  being  the  Naperian  base,  the  differential  of  y  =  e"^  is 

dy  =  ef'dx. 

II.    When  the  base  is  variable. 

Let  y  =  af.    Then  log?/  =  z  log  a;.     Differentiating  both  mem- 
bers, 

dy       dx      , 

-^  =  z h  logical, 

y       X 

whence       dy  =  x'z-^+  -^"^Jog  xdz  =  zx''^  dx  -{■  x'  log  xdz, 

or  the  differential  of  an  exponential  function  whose  base  is 
variable  is  the  sum  of  the  residts  obtained  by  differentiating  first 
as  if  the  exponent  were  constant  and  then  as  if  the  base  were 
constant. 

If  z  =  x,   y=xi',   and   rf^  =  af  (1 +loga;)  da*. 

Examples.     Differentiate : 

/7     ,1,,    /j*2 

1.  y  =  log (3 ax -f  x'^).  dy  =  3  — — — -dx. 


2.  y  =  loga^.  dy  = 

3.  y  =  (loga;)^  dy  =  2  log  x 

4.  ?/  =  log(loga;).  dy  = 

a-log  x 

5.  y  =  x  log  X.  dy  =  (log  a;  +  1 )  dx. 

•  y  =  \ — r  dy  = 


Zax  +  x^ 
2dx 

X 

dx 
x_ 
dx 


loga^  ""'  a;  (log  03^)^ 

7.  y  =  los(l-\-x^y,   or   21og(l  +  a^). 

ixdx 


dy  = 


l+ar« 


34 


THE   DIFFERENTIAL   CALCULUS. 
xdx 


S.  y  =  log  Vl  +  a^-  d}j  = 

d.  y  =  log  {Vl  +  x'  +  Vl  -  X-) 


1+a^ 


dx. 


10.  y 


:  log^-+^-^',    or   log  (1  +  -Vx)  -  log  (1  -  V.0- 


1- Va 


d!/  =  —= 


dx 


■\/x  (1  —  ic) 


11. 2/  = 


log[Vl-a;(l  +  a;)],    or   ^  log  (1  -  a;) +log  (1  +  a;), 
(1-Sx)dx 


12.  y 

13.  // 

14.  y 

15.  v/ 

16.  y 

17.  y 

18.  2/ 

19.  2/ 

20.  y 

21.  y 

22.  y 


=  log„4.TT. 

=  e^(l-cc-). 


dy  = 

dy  = 


2(1 -a^) 
mdx 

4:X 

dy  =  e^(l  —  2x  —  x^)dx. 
dy  —  af  (log  X  +  l)dx. 

dy  =  x^  e^  —L ^  dx. 

X 

dy  =  e^a;«^(log  x  +  l)dx. 


=  x^. 


.log  J 


dy  =  x^x 


log  a;  (logo; +  1)  4- 


dy  z=  2 x^"^'' log  X 


dx 


-.{logxy. 

=  log  {e"  —  e"). 


dy  =  (log  xy 


log  (log  x)  -f 


logcc 


dx. 


dx. 


dy 


dx. 


e'  —  e' 


e^  —  e' 


dy=  - 


4dx 


{e-e-'f 


dy  =  ('j  flog--l]dx. 


THE   TRANSCENDENTAL   FUNCTIONS.  35 

23.  y  =  (^.  dy  =  (fT(log^^  +  l\dx. 

Algebraic  functions  may  sometimes  be  differentiated 
with  greater  facility  by  first  passing  to  logarithms,  but 
it  is  usually  more  expeditious  to  differentiate  directly. 
Differentiate  the  following  by  passing  to  logarithms. 


24.  y  =  x^l-x{l+x). 

log  y  =  log  x  +  \  log(l  -  x)  +  log(l  +  x), 

y       \x     2{l-x)      1-^xJ       ' 

2  +  X-5.V' 


.-.  dy  =  icVl  —x(l-\-x\  — "^^^ ^-^ dx 

-^  ^  ^2x{l-x){\+x) 

=  1±^IzMdx. 


1  —  X  1  —x^ 

26.  y  =  ^•(^  +  ^).  dy  =  l+Mzil^i^d.. 

y/l-x"  ■(l-.x-)l 

27.  y  —  a^.  dy  =  (0"%"  log  a  log  hdx. 

28.  2/  =  J.  dy  =  ^!(ilzM^da.. 

29.  y  =  log        '^  ^    ~ dy=  - 


Vl  +  ic  —  Vl  —  X  a;  Vl  —  ix^ 

30.  y  =  a"''^\  dy=a}°^'' log  a—' 

dx 


31.  y  =  log( Va;  —  a  +  ^x  —  ?;).      f??/ 


2^{x-a){x-h) 


32.  ?/  =  log {x  —  Va;"  —  «-) .  dy—  — 


dx 


Vic'^  —  ci^ 


Qo  a;  ,        e'(l  —  x)  —  1  , 

33.  y  =  - — -.  dy=^— — -^  dx. 


36  THE   DIFFERENTIAL   CALCULUS. 

30,  Applications.  1.  Compare  the  rates  of  change  of  a  num- 
ber and  its  logarithm. 

a;  =  log„y,  whence  —  =  — ,   or  the  logarithm   (x)  changes 
dy      y 
faster  or  more  slowly  than  the  number  {y),  according  as  the 

number  is  less  or  greater  than  the  mq^ulus  of  the  system. 

Since  m  =  1  in  the  Naperian  system,  the  Naperian  logarithms 

of  proper  fractions  change  faster  than  the  fractions. 

2.  Compare  the  rates  of  change  of  a  number  and  its  loga- 
rithm in  the  common  system,  where  the  number  is  534.  The 
modulus  of  the  system  where  base  is  10  will  be  shown  to  be 

.434294,  .-.  -  =  1^5^?^  =  .00081,  which  will  be  found  by  ex- 

y  534 

amination  of  the  tables  to  be  the  tabular  difference  correspond- 

ing  to  the  number  534.     Since  —  changes  with  y,  the  relative 

rate  of  change  of  a  number  and  its  logarithm  varies  with  the 
number.  If  we  assume  that  for  an  increase  of  say  .1  in  the 
number  there  will  be  a  proportional  increase  in  the  logarithm, 
the  quantity  to  be  added  to  the  logarithm  of  534  to  obtain  the 
logarithm  of  534.1  will  be  .1  X  .00081  =  .000081.  This,  in  fact, 
is  the  manner  of  using  the  tabular  difference  of  the  tables, 

and  is  equivalent  to  the  supposition  that  —  remains  constant 

while  the  number  534  changes  to  534.1,  a  supposition  which, 
although  not  strictly  true,  gives  results  sufficiently  accurate 
within  the  limits  of  practice. 

3.  Find  the  tabular  difference  corresponding  to  the  number 
3217.  Ans.   .000135. 

4.  Prove  that  the  rule  for  the  differentiation  of  a  power 
applies  when  the  exponent  is  incommensurable. 

Let  y  =  a;",  n  being  incommensurable.   Passing  to  logarithms 

(first  squaring,  as  y  may  be  negative,  and  negative  numbers  have 

.  ,       ^    ,  ,  dy         dx 

no  logarithms),  log  y  =  n  log  x,  .-.  —  =  ?i  — ?  or  dy  =  ±  nx"  'ax. 

y         ^ 


THE   TRANSCENDENTAL   FUNCTIONS.  '        37 

5.  Prove  in  the  same  manner  that  the  rule  applies  when  the 
exponent  is  imaginary. 

6.  Find  the  slope  of  the  logarithmic  curve  at  the  point  where 
it  crosses  the  axis  of  Y. 

X  =  loga  y,  .:  -^  =  ~i  which  for  x  =  0  (whence  y  =  l)  becomes 

—     Since  y  =  1  when  a;  =  0,  whatever  the  base,  the  slopes  of 

all  logarithmic  curves  at  their  common  point  on  the  axis  of  Y 
vary  inversely  as  the  moduli  of  the  systems.  In  the  Naperian 
system  m  =  1,  hence  the  slope  of  x  =  log  y  is  the  ordinate  of 
the  point  of  contact. 

7.  Find  the  equation  of  the  tangent  to  x  -—  log  y. 

Ans.   y  —  y'  =  y'{x  —  x'). 

8.  Show  that  the  subtangent  of  a;  =  log^  y  is  constant  and 
equal  to  the  modulus  of  the  system.     Also  find  the  subnormal. 

.  ,dy'  ,  dx'      y"' 

Ans.    y'  ^—,  =  m:    v  3-,  = " — 
•^  dx'  '    ^  dy'      m 

9.  Compare  the  rates  of  change  of  x  and  its  ccth  power  when 
a;  =  1.  Ans.  The  rates  are  equal. 

10.  Compare  the  rates  of  change  of  x  and  its  iKth  root  when 

Ans.   ~-  =  0. 
dx 

The  Trkjonometric  Functions. 

31.   Circular  measure  of  an  angle. 

Any  angle  AOB,  measured  in  degrees,  may  also  be  measured 
by  the  ratio  of  its  arc  to  the  radius  of  its  arc,  since  for  any 
given  angle  this  ratio  is  constant  whatever 
the  radius  of  the  arc.     If  the  arc  6  be  de- 
scribed with  a  radius  equal   to   the   linear 

unit,  then,  since  x  =  r6  (Fig.  6),  -  =  B,  or, 

by  this  method,  the  angle  is  measured  by 

the  arc  intercepted  at  a  unit's  distance.    To  express  the  angle 


38  THE   DIFFERENTIAL   CALCULUS. 

n°  in  circular  measure,  we  have  -  = =  2  tt  for  the  circular 

r        r  2-77        IT 

measure  of  360° ;  hence  the  circular  measure  of  1°  is  — —  =  -— -, 

360      180 

and  of  71°  is  ^^ ;  or  the  circular  measure  of  an  angle  is  expressed 
180' 

by  multiplying  the  number  of  degrees  by  t^- 

180 

Since  -  =  1  when  x  =  r,  the  unit  of  circular  measure  is  the 
angle  whose  arc  equals  its  radius;  or,  making  — ^  =1,  n  = 

i-FTrt    r»  1  loO  IT 

=  57°.3  nearly. 

32.  Differential  of  since. 

Let  the  point  P  move  in  the  circular  path  AB,  x  being  the 
length  of  the  path,  estimated  from  A,  at  any  instant  when  the 
generating  point  is  at  P.     Then 

PD  =  y  =  sin  x. 

If  at  this  instant  the  motion  of  P  should  TV^g'?^' 

become  uniform  along  the  tangent  at  P,  the 
changes  in  AP  and  PD  would  also  become 
uniform.  Hence  if  PQ,  RQ,  are  what  the 
increments  of  x  and  y  would  be  in  any  interval  dt,  PQ  =  dx 
and  liQ  =  dy  =  d  (sin  x) .  But  BQ  =  PQ  cos  AOP.  Hence 
dy  =  cos  xdx,  or  the  differential  of  the  sine  of  an  angle  is  the 
cosine  of  the  angle  into  the  differential  of  the  angle. 

33.  Differential  of  cos  x. 

In  Fig.  7,  SD  =  BP,  being  the  decrement  of  OD  simultane- 
ous with  BQ  and  PQ,  is  the  differential  of  cos  x.     Hence,  if 
OD  =  y  =  cos  X,  dy  =  BP  =  —  PQ  sin  AOP=  —  sin  xdx. 
Otherwise :  ?/  =  cos  ic  =  Vl  —  sin^  x,  whence 

—  2sina:d(sin  x)  sin  aj  cos  a^dx  .       , 

dv  = ^         -  =  — =  —  sm  xdx, 

2Vl-sin2aj  cosa;      . 

or  the  differential  of  the  cosine  of  an  angle  is  minus  the  sine  of  the 
angle  into  the  differential  of  the  angle. 


THE   TRANSCENDENTAL   FUNCTIOl^S.  39 


34.   Differential  of  tan  x. 

sin  X      -_,, 

Let  V  =  tan  x  = inen 

^  cos  a; 


,        cos  a;d(sinx)  — sin.Trf(cos.r)      cos'a; -l-sin'^a;  , 

dy  = 5^ '— ^ '-  = \ dx 

Q,Q'S>^x  COS'' a; 


dx         _ .  2 


cos^a; 


sec^a^dx, 


or  the  differential  of  the  tangent  of  an  angle  is  the  square  of  the 
secant  of  the  angle  into  the  differential  of  the  angle. 

35.  Differential  of  cot  x. 

Let  y  =  cot  x  =  tan  [  ^  —  re  j .     Then 

dy  =  sec^ I-  —  x\{  —  dx)  =  —  cosec^ xdx,  a  result  whicli  may 

•  COS  x 

also  be  obtained  by  differentiating  y  =  cot  x  = Hence 

sin  X 

The  differential  of  the  cotangent  of  an  angle  is  minus  the  square 
of  the  cosecant  of  the  angle  into  the  differential  of  the  angle. 

36.  Differential  of  sec  x. 

Let  y  =  sec  x  = Then 

COSiC 

,  d(cosa;)      sina/'d.«  ,  , 

dy= ^^ — - — -  =  ——, —  =  sec  x  tan  xdx, 

cos-  X         cos-  X 

or  the  differential  of  the  secant  of  an  angle  is  the  secant  of  the 
angle  into  the  tangent  of  the  angle  into  the  differential  of  the  angle. 

37.  Differential  of  cosec  x. 

Let  y  =  cosec x  =  sec (-  —  x).     Then 
V2        J 

dy  =  sec  [  ^  —  x\  tan[  -  —  a;  j  ( —  dx)  =  —  cosec  x  cot  xdx, 

or  the  differential  of  the  cosecant  of  an  angle  is  minus  the  cosecant 
of  the  angle  into  the  cotangent  of  the  angle  into  the  differential  of 


40  THE  DIFFERENTIAL   CALCULUS. 

38.  Differential  of  vers  x. 

Let  y  =  vers  x=l  —  cos  x.     Then  dij  =  sin xdx, 

or  the  differential  of  the  versine  of  an  angle  is  the  sine  of  the  angle 
into  the  differential  of  the  angle. 

39.  Differential  of  covers  x. 

Let  y  =  covers  x  =1  —  sin  x.     Then  dy  =  —  cos  xdx, 

or  the  differential  of  the  coversine  of  an  angle  is  minus  the  cosine 
of  the  angle  into  the  differential  of  the  angle. 

Examples.     Differentiate : 

1.  y  =  sin  6  x.  dy  =  6  cos  6  xdx. 

2.  y  =  cos  XT.  dy  =  —  2x  sin  x-dx. 
y  3.  y  =  cos^  x^                dy  =  —  sin  2  xdx. 

4.  ?/  =  tan  (3  -  5  x^) \     dy  =  -  20  a; (3  -  5  x")  ^ec\3  -  o  x^f  dx. 

5.  y  =  sin2  ^^  _  2x^y-.     dy  =  -  8a;(l-  2ic2)sin  2(1-  2x'y-dx. 
^    6.  2/  =  (sin  a;  cos  x)  ^       dy  =  sin  2  a;  cos  2  xdx. 

'^  _  7.  2/ =  sin  2 a:  cos  2 a;.       dy  =  2  cos  4:xdx. 

'^  S.  y  =  sin2  (1  -  a.-^)^.       dy  =  -Sx{l-x-)  sin  (1  -  x^dx.  V 

1      ^          ^          .                   J        1  +  sin  a;  J 
'      9.  V  =  tan  X  +  sec  x.       dy  =  — '—^ dx. 

COS''  X 

10.  y  =  X  -j-  sin  x  cos  x.    dy  =  2  cos^  xdx. 

O^ll.  y  =  tanVl  — a;-.       .  dy  =  —  sec^  Vl  -  x^  • 

\.  Vl  —  x^ 

-'-^'-                      ,                      ,       cos  (log  x)  , 
^.12.  y  =  sin  (log  a;) .  dy  = ^^-^^  dx. 

'''da; 

13.  2/  =  log  (cot  a;).  dy  =  --^-^. 

14.  y  =  m  sin"  ax.  d?/  =  a??m  sin"  "^  ax  cos  axda;. 

15.  t/  =  sin'a;.  dy  =  sin''a;(log  sin  a;  4-x  cot  a;)  dx. 


THE   TRAXSCENDENTAL   FITNOTIONS.  41 

16.  ?/  =  vers  -•  dy  =  -  sin  -  dx. 

17.  y  =  sin  e"^.  d?/  =  e^  cos  e'  dx. 

18.  ?/  =  .^•^  cos  0/*^.  dy  =  2  a; (cos  cc-  —  a^  sin  a;-)  d;f. 

io  •    "  7  tt         d  , 

19.  w  =  sin  -  •  dw  = -„  cos  -  da\ 

^  X  -^  x^        X 

20.  y  =  log  (sin  a-) ,  d//  =  cot  xda;. 

21.  _?/  =  sin  «a;  sin''a;.  dy  =  a  sin"~  '.'c  sin  (ax  +  x)  dx. 

J.  1, 

oo  4-        X  7  sec- a*  log  a  •  a*  da; 

22.  y  =  tn,nce.  dy  = ^ • 


ar 


23.  y=zx'"'\  dy  =  a; »'" *  /sin^  _,_  j^g  ^j  cos  a;"]  da;. 

24.  y  =  (sina;)'"°". 

dy  =  (sin  a;)™**  |cot  x  cos  a;  —  sin  x  log  sin  a;  j  d-r. 

(sin  na;)"*  ,       ?nn(sin na;)'""^ cos (mx— ?ia;)  , 

2i>.  y  =  -7 T7-  d?/  = ^^ ^ r-xi -dx. 

^      (cos  ma;)"  "^  (cosmx)"+^ 

The  Circidar  Functions. 

40.  Differential  of  sin~'ic. 

Let  y  =  sin~^T.     Then  x  =  sin  y.     But  dx  =  cos  ydy,  hence 

d   _  ^^    _  da;         _       dx 

~cosy~  Vl-sin^y  ~  VH^^' 

or  i/te  differential  of  an  arc  in  terms  of  its  sine  is  the  differential 
of  the  sine,  divided  by  the  square  root  of  1  minus  the  square  of 
the  sine. 

41.  Differential  of  COS" ^aj. 

Let  y  =  cos^a;.    Then  x  =  cos  y.     But  da;  =  —  sin  ydy,  hence 
,   _  _    f?.i7   _  da;         _  _       dx 

si"  y  Vl  -  cos^y  Vl-a;' 


42  THE   DIFFERENTIAL   CALCULITS. 

or  the  differential  of  an  arc  in  terms  of  its  cosine  is  minus  the 
differential  of  the  cosine,  divided  by  the  square  root  of  1  minus 
the  square  of  the  cosine. 

42.  Differential  of  tan^^a?. 

Let  y  =  t'An~^cc.     Then  x  =  timy.     But  dx  =  sec- ydy,  hence 

,  dx  dx  dx 

(^y  =  — V-  =  rTT — ~  =  1— , — ■>' 
sec^y      1  +  tan-  y      1  +  ar 

or  the  differential  of  an  arc  in  terms  of  its  tangent  is  the  differen- 
tial of  the  tangent,  divided  by  1  plus  the  square  of  the  tangent. 

43.  Differential  of  cot"'  x. 

Let  y  =  cot~^  x.       Then  x  —  cot  y.       But   dx  =  —  cosec''  ydy, 

hence 

7    _    ~  '^^•^"   _  _        '^^^       _  _     ^^ 
cosec-  y  1  +  cot-  y         l-\-x^ 

or  the  differential  of  an  arc  in  terms  of  its  cotangent  is  minus  the 
differenticd  of  the  cotangent,  divided  by  1  pilus  the  square  of  the 
cotangent. 

44.  Differential  of  sec  ~^  35. 

Let  y  =  sec~^  x.      Then   x  =  sec  y.      But   dx  =  sec  y  tan  ydy, 

hence 

dx  dx  dx 

dy  = 


sec  y  tan  y      gee  2/ Vsec- ?/  -  1      re  Va^  -  1 

or  the  differential  of  an  arc  in  terms  of  its  secant  is  the  differential 
of  the  secant,  divided  by  the  secant  into  the  sqxiare  root  of  the 
square  of  the  secant  minus  1. 

45.   Differential  of  cosec  "^£c. 

Let  ?/=cosec~^a;.  Then  a; = cosec y.  But  c?a7=— coseca;cota;(7x, 

lience  ^^^ 

dri  = > 


THE  TRANSCENDENTAL   FUNCTIONS.  43 

or  the  differential  of  an  arc  in  terms  of  its  cosecant  is  minus  the 
differential  of  the  cosecant,  divided  by  the  cosecant  into  the  square 
root  of  the  square  of  the  cosecant  minus  1. 

46.   Differential  of  vers"^ic. 

Let  y  =  vers'^cc.     Then  x  =  vers  y.    But  dx  =  sin  ydy,  hence 

,          dx  dx  dx 

dy  =  -, —  = 


sin  y      -y/i  _  cos^ y      Vl  —  (1  —  vers  y)'^ 
dx  dx 


or  the  differential  of  an  arc  in  terms  of  its  versine  is  the  differential 
of  the  versine^  divided  by  the  square  root  of  twice  the  versine  minus 
the  sqxiare  of  the  versine. 

47.  Differential  of  covers"^  a?. 

Let  y  =  covers"' x.    Then  x  =  covers  y.     But  dx  =  —  cos  ydy, 

hence  , 

,  dx 

dy=- 


V2x'-x2 

or  the  differential  of  an  arc  in  terms  of  its  coversine  is  minus  the 
differential  of  the  coversine,  divided  by  the  square  root  of  tivice 
the  coversine  minus  the  sqxiare  of  the  coversine. 

Examples.    Differentiate : 

^xdx 


VI -4  a;* 
dx 


VI 


1.  y  =  sin~' 2.x*^.  dy  = 

2.  y  =  cos  'Vl  —  a^.  dy  = 

S.  V  =  sin~'  -^ —  dy  = 
^              1  +  a^  ^  1+x 

1 

4.  y  =  tan  'a*.  dy  =  — 


X- 
2dx 


1 
a^  lo»  adx 


x" 


(l  -j-  a') 


44  THE   DIFFERENTIAL   CALCULUS. 


5.  2/  =  tan-V.  dy  = 


dx 


e^  +  e' 


6.  2/  =  sin-i(tancc).  dy  =  - ^^^-^^~ ~  . 

Vl  —  tan''  X 

7.  y  =  cos-\2  cos  x).  dy  = ?smxdx     ^ 

Vl  —  4oos^x 


S.  y  =  cos'^(]ogx).  dy=  — 

9.  ^  =  log(cos-ia;).  dy  = — 


dx 


Vl  —  log^  a; 


cos  Iojy'I  — a^ 

10.  2/  =  tan--^.  dy  =  ^(^-^)^^. 

1+a^  ^      l  +  (jx'  +  x* 

11.  2/  =  a?  sin-i  X  -  Vl  -  x^.         dy=(sm-^x-\ ~ — \dx. 

\  Vl  -  x'j 

12.  x  =  r  versin"^^  —  V2  rv  —  v^ 

dx=    y^y    . 

V2  r?/  -  2/2 

13.  y  =  (sin~^a;)^ 

f?.y  =  (sin  'a;)-'  j  sin-^a^logCsm-^a;)  Vl  -  a;''  + a; )  ^^^ 
<.  Vl  -  a.-^  > 

14.  2/  =  a?"""'^  ^2/  =  ^^"""■'K  I  ''^^"'^  +    ^Qg^- 1  dx 

i      X  Vl  -  af'  ^ 

15.  2/  =  siu-i-.  dy  =  —^^~-. 
IG.  2/  =  cos-i?.                              d2/=  '^''' 


'■  Vr^  -  ar* 

a'  .7  ?TZa; 


17.  2/  =  tan-i--  c?2/  = 


5.2  _|_  ^2 


18.  2/  =  cot-i^.  dv=-    ^'^^  . 

r  ^  y~-  ^_  ar* 


THE  TRANSCENDENTAL  FUNCTIONS.  45 


19.  y  =  sec    -•  cly  = 


'>'  x^ar 


r 


20.  y  =  cosec"''—  dy=  — 


r  a;  Va^  - 

21.  y  =  vers~^—  dy  = 


'•  -^2rx  —  cc 

22.  1'/  =  covers"^ -•  dy= — 


>'  -yj^rx-^ 


23.  2/ =  tan-\ -— -^^^.  dy  =  \dx. 

\  1  +  cos  X' 


24.  2/  =  sin~^ Vsm^.  dy  =  ^Vl  +  cosec  x  dx. 

25.  2/  =  log('^y4-itan-^x-.   ^^2/  =  ^,- 

26.  ?/  =  sec~'- 


dx 


27.  2/  =  sin-»?-±^ 


+1 

V2 


cZ?/  = 

vr 

—  af' 

d?/  = 

dx 

vr 

-2x- 

-x" 

dt/  = 

(^20; 

tan-i^ 
r 

+  r 

^da;. 

rfy  = 

ndx 

cos^ 

x-i-n^i 

sin- 

a; 

dy  = 

-2dx. 

28.  2/  =  0-2-|-a^)tan-^-. 

?' 

29.  y  =  tan~^(9itana;). 

30.  ?/  =  cos~'(cos2a;). 

48.  Applications.  1.  A  wheel  revolves  about  a  fixed  axis 
through  its  centre.  Compare  the  velocity  of  a  point  on  the 
rim  with  its  velocity  in  a  horizontal  direction. 

The  horizontal  velocity  is  evidently  the  rate  of  change  of  the 
cosine  of  the  arc  described  by  the  point ;  hence,  if  the  arc  be 
denoted  by  «,  y  =  cos  x,  whence  dy=  —  sin  xdx,  which  is  also 
the  relation  between  the  rates  of  y  and  x.  The  point  is  there- 
fore moving  in  a  horizontal  direction  sin  x  times  as  fast  as  it 


46 


THE   DIFFERENTIAL   CALCULUS. 


is  moving  in  the  arc.  At  the  highest  point,  where  x  =  90°, 
sina;=l,  and  dy  =  —  dx,  the  rates  being  equal.  At  a;  =  30°, 
sin  x  =  ^,  and  at  this  point  the  horizontal  velocity  is  one-half 
that  in  the  arc. 

2.  Compare  the  horizontal  and  vertical  velocities  of  a  point 
on  the  rim  of  a  wheel  which  rolls  without  sliding  with  a  constant 
velocity  m  on  a  horizontal  line. 

In  this  case  the  path  of  a  point 
on  the  rim  is  a  cycloid  whose  equa- 

-1         V2?-?/-2/^, 


tion  IS   X  =  r  vers' 
whence  —  = — 


dy 


,fidt 


(!)• 


ON 


dt      -^2  ry  —  y 
Since  the  wheel  has  a  constant 
velocity  m  in  a  horizontal  direction,  and  its  centre  C  is  always 
vertically  over  Z>,  this  velocity  is  the  rate  of  change  of 

OD  =  r  vers~^  -  • 


d 


Hence 


r vers  '  - 
r 


dt 


V2ry 


dy 


Substituting  this  value  in  (1), 

dx     y 

— -  =  -  m. 
dt      r 


Hence 
At  O, 
At  B, 

At  E, 


'^y 


ds         l/dxY     fdyV  /2i 

dt-\{dt)+[i)  =  '^\i- 

dy 


y  =  0,     and 


dx 
dt 


=  ^  =  0. 
dt      dt 


^   dx     ds      ^      dy      ^ 
2/  =  2r,  and  --  =  -  =  2m,^  =  0. 


dt      dt 


dt 
ds 


^   dx      dy 
y^r,     and  -  =  ^-  =  m,^^ 


tV2. 


y 


I, 

f)  THE   TRANSCENDENTAL   FUNCTIONS.  47 

3.  Find  the  subnormal  of  the  cycloid. 

y''^  =  y'  V27y-y;^  ^  ■V2i'y'-y'-\  But  (Fig.  8)  ■V2ry'-y'-' 

=  PM  =  ND,  or  the  normal  passes  through  the  foot  of  the 
vertical  diameter  of  the  circle  when  in  position  for  the  point 
P.  Hence,  also,  the  tangent  passes  through  the  upper  extrem- 
ity T.  Therefore  to  draw  a  tangent  and  a  normal  at  any  point 
P,  put  the  generating  circle  in  position  and  join  P  with  the 
extremities  of  its  vertical  diameter.  Also,  to  draw  a  tangent 
parallel  to  a  given  line,  draw  BQ  parallel  to  the  given  line,  and 
PQ  parallel  to  the  base.  Then  P  is  the  required  point  of  tan- 
gency. 

4.  A  man  walks  in  a  direction  AB.  Compare  the  rate  of 
change  of  his  distance  from  a  point  0  with  the  rate  of  his 
angular  motion  about  0. 

Let  fall  the  perpendicular  0Z>  =  2:>  upon  AB,  and  take  0  for 
the  pole,  OD  for  the  polar  axis.     Then  the  equation  of  AB  is 

p        ■  ,  dr      p  sin  $  d6       ^       /,      a     f^**     a      r 

r=-^,  whence  ^t=- — ^r?r -rr-      For    ^  =  0,    -,^  =  0;    for 
cos  6  dt       cos- 6  dt  '    dt        ' 

6  =  90°,  -  =  00. 
dt 

5.  An  elliptical  cam  making  two  revolutions  a  second  about 
a  horizontal  axis  through  one  focus,  gives  motion  to  a  bar  in  a 
vertical  direction  through  the  centre  of  revolution.  The  trans- 
verse axis  being  G  and  the  eccentricity  f,  find  the  velocity  of 
the  bar  when  the  angle  between  the  vertical  and  the  trans- 
verse axis  is  60° ;  90°. 

a{l  —  e-)        ,  dr         a(l  —  e-)esmO  dd      ...    „ 

r  =  z ~,  whence  -j-  = yz — jr-r-  -77'  which  for 

1— ecos^  dt  {1—ecos  6y   dt 

a  =  3,  e=|,  and  —  =  47r,  becomes -tt.    When 

^  dt  '  {S-2cosOy 

0  =  60°,  ^=-5V37r;  when  0  =  90°,  ^'  =  -12^. 
dt  dt  9 

6.  The  crank  of  a  steam  engine  is  one  foot  in  length  and 
makes  two  revolutions  a  second.     If  the  connecting  rod  is  5 


48  THE   DIFFERENTIAL   CALCULUS. 

feet  in  length,  find  the  velocity  of  the  piston  when  the  crank 
makes  angles  of  45°,  135°,  90°,  with  the  line  of  motion  of  the 
piston  rod.  Let  a,  b,  x,  represent  the  crank,  connecting  rod, 
and  variable  side  of  the  triangle,  respectively,  and  6  the  angle 
between  a  and  x.     Then  x  =  a  cos  $  +  V&"  —  a^sin^  6,  whence 

dx           (       ■    a  ,     a^  sin  0  cos  0    ]  dO 
—  =  —  a  sm  6  H —  [  —  ? 

dB 
which  for  a  =  1,  ?>  =  5,  —  =  47r,  becomes 
dt 


f    .    /)  ,     sin  6  cos  6    ) 
\  sin  ^  H -  y 

(  ■\/25  -  sin2  0  ) 


V25  -  sin^  6 

Ans.    -if-^Tr;   --y_V27r;   -47r. 

7.  Find  the  slope  of  ?/  =  sin  x  at  the  points  where  the  curve 
crosses  X.  Ans.    ±  1. 

8.  Find  the  angle  at  which  y  =  sin  x  crosses  y  =  cos  x. 

Ans.   tan-i2V2. 

9.  Find  the  length  of  the  normal  to  the  cycloid. 

Ans.    -y/'^ry'. 


CHAPTER   III. 

SUCCESSIVE   DIFFERENTIATION. 

49.  Equicrescent  variable.  A  variable  lohich  changes  uni- 
formlJ^,  fTiat  is,  whose  rate  is  constant,  is  said  to  be  equicrescent. 

50.  The  differential  of  an  equicrescent  variable  is  constant. 

For,  if  X  be  equicrescent,  its  rate  -^  is  constant.     But  dt  is 

constant ;  hence  dx  is  also  constant. 

It  is  evident  that,  if  —  is  not  constant,  dx  is  a  variable. 
dt 

The  above  is  a  direct  consequence  of  the  definitions  ;  for  the  differen- 
tial of  a  variable  is  what  would  be  its  change  during  any  interval  were  its 
rate  of  change  to  remain  throughout  the  interval  what  it  was  at  its  begin- 
ning. If  the  rate  varies  from  instant  to  instant,  differentials  correspond- 
ing to  equal  intervals  also  vary;  while  if  the  rate  remains  the  same,  these 
differentials  are  equal. 

51.  Successive  derived  equations. 

Let  y=f(x).  Then  d>j=f'(x)dx,  in  which  f'{x)=-^, 
the  first  derivative. 

Now,  in  general,  dy,  or  /'  {x)  dx,  is  a  variable.  For  dx  is  a 
variable  unless  x  is  equicrescent ;  and  /'  (x)  is  a  variable  unless 
f{x)  is  linear,  in  which  case  it  can  be  reduced  to  the  form 

dv 
y  =  mx  -f  b,  whence  -r-=f'  (x)  =  m,  a  constant.     Hence,  unless 

the  function  is  linear  and  x  is  equicrescent,  dy  =f'(x)dx  is 
variable,  and,  being  true  for  all  values  of  x,  can  be  differenti- 
ated, thus  forming  a  second  derived  equation  which  may  in  its 
turn  be  differentiated,  a  repetition  of  this  process  leading  to 

49 


50  THE   DIFFERENTIAL   CALCULUS. 

successive  derived  equations  called  the  fii^t,  second,  third,  etc., 
in  order. 

Since  differentiation  introduces  no  function  which  has  not 
been  already  treated,  the  successive  derived  equations  are  ob- 
tained by  the  rules  already  established. 

52.  Notation.  The  second  differential  of  a  variable  x  is 
represented  by  the  symbol  cBx,  read  '  second  differential  of  a;,' 
the  exponent  being  a  symbol  of  operation  indicating  how  many 
times  the  variable  has  been  differentiated.  The  student  will 
observe  the  different  meanings  of  the  forms  cZ-a;,  doi?,  and  d  (x')l 

Illustration.  Given  if  =  2px.  The  first  derived  equation 
is  2ydy  =  2pdx,  or  ydy  =  pdx.     Differentiating  again,  we  have 

yd{dy)  +  dyd{y)  =pd  {dx), 
or,  in  the  above  notation, 

yd-y  +  dy^  =  pd-x, 
which  is  the  second  derived  equation.     Differentiating  again, 

yd{d'y)  +  dhjd{y)  -f  2  dyd{dy)  =pd{d'x), 
or  yd^y  +  d-ydy  +  2  dydnj  =  pcZ%, 

whence  yd^y  -\-  3  dyd^y  =^?d''a;, 

which  is  the  third  derived  equation. 

If  X  were  equicrescent,  the  successive  derived  equations 
would  be  much  simplified.  For  when  x  is  equicrescent,  dx  is 
constant,  and,  since  the  differential  of  a  constant  is  zero,  all 
the  successive  differentials  of  x  after  the  first  would  vanish. 
Thus,  in  the  above  illustration,  d'x  =  d^x  =  etc.  =  0,  and  the 
successive  derived  equations  become 

ydy  =  2^dx, 
yd^y  +  dy^  =  0, 
yd'y  +  3dyd'y  =  0. 

53.  Remark.  It  is  important  to  observe  that  in  most  cases 
it  is  permissible  to  consider  the  variable  equicrescent  and  thus 


SUCCESSIVE   DIFFERENTIATIOX.  51 

secure  the  simplicity  above  noted.  For  example,  let  y  =f{x) 
be  the  equation  of  any  plane  curve.     The  assumption  that  x  is 

equicrescent,  or  that  —  is  constant,  implies  that  the  velocity 

of  the  generating  point  in  the  direction  of  the  axis  of  X  is  con- 
stant. Now,  so  far  as  the  geometrical  properties  of  the  curve 
are  concerned,  these  being  independent  of  the  velocity  of  the 
generating  point,  we  are  at  liberty  to  make  any  assumption  re- 
garding the  velocity  which  will  facilitate  their  investigation. 
We  therefore  assume  the  velocity-law  in  the  curve  such  that 
the  motion  in  the  direction  of  the  axis  of  X  is  uniform. 

Again :  suppose  a  right  cylinder  is  inscribed  in  a  right  cone, 
the  problem  being  to  find,  of  all  right  cylinders  so  inscribed, 
that  one  whose  volume  is  the  greatest.  If  the  radius  of  the 
base  and  altitude  of  the  cone  are  h  and  a,  and  those  of  the 
cylinder  x  and  y,  we  have 

h  :  a  : :  X  :  a  —  y, 

whence  x  =  -  {a  —  y)\ 

and  if  V  is  the  volume  of  the  cylinder, 

F=  iryxr  =  TT  -  y{a  -  yy. 
cr 

Now  in  determining  the  gred,test  value  of  V,  it  is  evidently 
immaterial  whether  we  regard  y  equicrescent  or  not,  since  the 
cylinder  of  greatest  volume  is  independent  of  the  law  of  change 
oiy. 

In  functions  of  a  single  variable,  unless  mention  is  made  to  the 
contrary,  the  variable  will  hereafter  be  regarded  equicrescent. 

Examples.     Regarding  x  equicrescent,  form : 
1.  The  second  derived  equations  of 

aV  +  Wa?  =  a^S  a-yd^y  +  a-df-  +  b'dx^  =  0. 

f  +  x^  =  R^,  yd?y  +  df'  +  daf  =  0. 

xy  =  m,  2  dydx  -f  xd^y  =  0. 

y  =x^  log  X,  d^y  =  2  log  xdx  -f-  3  dx^. 


A 


52  THE   DIFFERENTIAL   CALCULUS. 


2.  The  fifth  derived  equation  of  y  —  x^  log  x. 

24 
d^y  =  —  dx^. 

X 


3.  The  fourth  derived  equation  oi  y  = 


X 


\-x 


24 
d'y  =  —^^^±—dx\ 

4.  The  third  derived  equations  of  : 

y  =  tan  x,  d^y  =  2(3  sec^  x  —  2)  sec^  xdx\ 

y  =  e*,  d'^y  =  e'^da;^. 

y  =  -,  d^y  = dx\ 

X  X* 

y  =  COS  X,  dhj  =  sin  xdx\ 

5.  Prove  that  the  lith  derived  equation  of  y  =  a'  is 

d"y  =  (log  a)"a'(te". 

6.  If  2/  =  log  sin  X,  prove  that    d^y  =  2 darl 

sin^a; 

7.  If  y  =  sin~^Va;,  prove  that    d-y  = ^ dx^. 

4{x-x'y^ 

8.  If  y  =  m  cos^mcc,  prove  that 

d^y  =  —m*\  (cos  mx')"*  —  (m  —  1)  (cos  wa;)"*"^ sin^  mxldaf. 

9.  If  2/  =  a«^,  show  that  d^y  =  0. 

54.   Successive  derivatives,  or  differential  coefficients. 

Let  y=f{x),  in  which  x  is  equicrescent.    The  first  deriva- 

dv      d,\  f(x)~\ 
tive  oif{x)  has  been  defined  as  -j-=    "-^  ^,  and  is  the  ratio 

of  the  rates  of  change  of  the  function  and  its  variable.  Since 
the  first  derivative  is  variable  except  when  f(x)  is  linear,  it  is 
in  general  a  function  of  x  and  may  be  denoted  by  f'(x),  or 

dv 

—  =f'(x)  ;  it  may  therefore  be  differentiated  in  its  turn,  and 

a  second  derivative  formed  by  dividing  d\_f'{x)']  by  dx,  and 


SUCCESSIVE   DIFFERENTIATION.  53 

this  process  may  evidently  be  continued  until  a  derivative  is 
reached  which  is  constant.  The  successive  derivatives  thus 
obtained  are  called  in  order  the  first,  second,  third,  etc.,  deriva- 
tives, and  are  denoted  hj  f'(x),f"(x),f"'(x),  etc. 

Since  each  derivative  is  obtained  from  the  preceding  one  in 
the  same  manner  that  f'(x)  is  obtained  from  f(x),  it  follows 
that: 

1.  The  nth  derivative  of  f{x)  is  the  ratio  of  the  rate  of  change 
of  the  (n  —  l)th  derivative  to  that  of  the  variable. 

2.  The  nth  derivative  off{x)  may  be  obtained  either  by  differ- 
entiating the  (n  —  l)th  derivative  and  dividing  by  dx,  or  by  divid- 
ing the  nth  derived  equation  by  dx". 

Illustratiost.     Given  y  =  a  -j-  bx^.     The  first  derivative  is 

^y  =  Sba^=f'(x).^-^^  "■''    •  ^*"    ''' 

dx  J  \   / 

Differentiating,  remembering  that  dx  is  constant, 

— ^  =  6  bxdx, 
dx  ' 

whence  the  second  derivative 

g  =  6to=/"(x).  •  v: 

Differentiating  again, 

d?v 
^=Gbdx, 


whence  the  third  derivative 

cPy 


dx" 


z=6b=f"'{x).-Uc^.  U-i{  M  ^i '    ^^ 


Here  the  process  ends,  since  the  third  derivative  is  constant. 
Otherwise,  differentiating  y  =  a+  ba?  successively  three  times, 
the  successive  derived  equations  are 

dy  =  3  bx^dx,  d^y  =  6  bxdx^,  d^y  —  6  bda?, 


54  THE  DIFFERENTIAL   CALCULUS. 

and,  dividing  the  last  by  dx"^, 

as  before.  •  "^ 

55.  Sign  of  the  nth  derivative.  Since  f'(x)  is  positive  or 
negative  as  f{x)  is  an  increasing  or  decreasing  function  (Art. 
22),  and  since /"(x)  is  the  first  derivative  of  f'(x),  f"'(x)  the 
first  derivative  of/"(aj),  etc.,  therefore /"(a;)  will  be  positive  or 
negative  asf"~^{x)  is  an  increasing  or  a  decreasing  function. 

Examples.    1.  If  y  =  mx"',  prove  that  ~,  ovf"'(x),  is 

m^(m  —  1)  (m  —  2)  03""^ 

f'(x)  =m^a;'"~\ 
f"{x)  =  m^{m  —  1)0^-^ 
f"'lx)  =  m\m-l){m-2)x"'-^ 

2.  If  2/  =  e*  sm  x,  prove  that  — ^  =  2  e*  cos  x. 

f'(x)  =  e*  cos  x-^-e"  sin  x  =  e''  (cos  x  +  sin  a;) . 
f"(x)  =  e''(—  sin  x  +  cos  cc)  +  e"'(cos  x  -f  sin  ic)  =2e'  cos  ic. 

3.  If  2/  =  log  cos  X,  prove  that  /'"(x)  =—2  sec^ic  (3  sec^a;  —  2). 

4.  If  2/  =  Vl  —  ar^,  prove  that  —  = • 

dx-         if 

5.  li  y  =  e""'',  prove  that 

f"'{x)  =  e^^'^'Q.o&x  (cos^  a;  —  3  sin  a;  —  1). 

6.  If  y"^  =  sec  2  x,  prove  that  /"(a;)  =  3y^  —  y. 

7.  If  y  =  a''^,  prove  that  f\x)  =  h^  log^a  •  a*^ 

8.  If  ?/^  =  2px,  and  y  is  equicrescent,  prove  that  —  =  -  • 

dy^     p 

The  following  first  and  second  derivatives,  being  of  frequent 
use  hereafter,  may  be  here  established  for  future  reference. 
In  all  implicit  functions  of  two  variables,  x  will  be  regarded  as 
the  equicrescent  variable  unless  otherwise  mentioned. 


SUCCESSIVE   DIFFERENTIATION.  55 


9.  The  ellipse,  a-y'^+  &V=  a-b-. 

10.  The  circle,  ^f+x'=  R-. 

f{x)  =  -^=T     ,   ^ •  f\x)  =  -^. 

11.  The  hyperbola,  a^  —  Ira?  =  —  a~b^. 

/'(a.)=^=±^-^==:.         /"(x)=— ^. 

12.  The  hyperbola,  ir?/  =  m. 

a;         a-  ir        a^ 

13.  The  parabola,  y'^  =  2px. 

/'(..) =?;=±-^.  /"(a.) =-4 

14.  The  cubical  parabola,  'f  =  o?x. 

f(x\--^--^.  f"(x\-     2^' 

15.  The  semi-cubical  parabola,  ay''  =  ar\ 

2a?/  2  \a  4tty 

16.  The  witch,  x^y  =  i  a\2  a  -  ij) . 

fi(^\^_     2a7/  16  a"  a. 


x2  +  4a-  (iK2-j-4a2)2 

f"(x)  =  2y  ''^^'~'^f„ 

w  

17.  The  cycloid,  x  =  r  versin"^-  —  V2?'?/  —y-. 


66  THE   DIFFERENTIAL   CALCULUS. 

18.  The  cissoid,  ->/  =  — 

2a  — X 

f'(x)  =  ±  Xi       3^^-^  /"  (,;)  =  ±  1^ 

{2a -x)^-  x'-{2a-xy^ 

19.  The  hypocycloid,  x^  -f  y^  =  a*. 

1  .  2 

/'(^•)  =  -V  /"(^)=o-r~4- 

a:  X 

20.  The  catenary,  v/  =  -  (e"  +  e  "). 

^  ft 

21.  The  logarithmic  curve,  x  =  logy. 

f'(x)=f"(x)  =  y. 

22.  The  sinusoid,  y  =  sin  a;. 

/'(a;)  =  coscc.  f"(x)  =  —  su\x  =  —  y. 

56.  Remark.  If  a  function  becomes  infinite  for  ajinite  value 
of  the  variable,  its  derived  functions  also  become  infinite. 

For  if  the  function  be  an  algebraic  one,  it  can  become  infinite 

for  a  finite  value  of  the  variable  only  by  having  the  form  of  a 

fraction  whose  denominator  vanishes  for  that  value,  and,  in 

differentiating  to  form  the  derived  functions,  this  denominator 

never  disappears.      So  that  if  f{x)  =  co  when  x  =  x',f'{x), 

f"{x),  etc.,  also  become  infinity  when  x  =  x'.     Examination  of 

the  transcendental  functions  leads   to   the   same  conclusion. 

Thus  log  X  becomes  infinity  when  x  =  0,  as  do  also  all  its  deriv- 

11  - 

atives  -, -,  etc. :  and  a'',  tan  x,  sec  x,  illustrate  the  same  fact. 

X  XT 

This  is  not  necessarily  true  when  f{x)  becomes  infinity  for 
an  mfinite  value  of  the  variable.    Thus,  log  a?  =  oo  when  a;  =  oo  ; 

but  f\x)  =  -  becomes  zero  for  x=  cc. 


SUCCESSIVE   DIFFERENTIATION.  67 

57.  Notation.  To  denote  what  a  function  becomes  for  a 
particular  value  of  the  variable,  the  variable  is  replaced  by  its 
particular  value.  Thus,  /(a),  /(O),  f{x'),  represent  what  f(x) 
becomes  when  x  =  a,  a*  =  0,  x  =  x',  respectively.  The  particu- 
lar value  may  also  be  written  as  a  subscript  in  either  of  the 
following  ways : 

1]        =0,     1]    =0, 

XJ  a;  =  CO  XJ  ^ 

( 1 
read     -  equals  zero  when  x  is  infinity.' 

58.  Change  of  the  equicrescent  variable. 

In  forming  the  successive  derivatives  of  y  =f{x)  wo  have 
considered  x  equicrescent,  that  is,  dx  constant,  and  hence 
drx  =  cfx  =  etc.  =  0. 

If  x  is  not  equicrescent,  dx  is  a  variable,  and 

am 

\dxj      dxd-y  —  dy(Px 


dx  dx^ 


(1) 


which  is  the  general  form  of  the  second  derivative  when  neither 
x  nor  y  is  equicrescent. 

Differentiating  (1),  regarding  dx  and  dy  as  variables,  we  have 

fdxd^y  —  dyd-x\ 

\  daf         J      ((f'ydx  —  d^xdy)dx  —  3(d'ydx  —  d?xdy)d?x 


dx  dx  x^v 

which  is  the  general  form  of  the  third  derivative  when  neither 
X  nor  y  is  equicrescent.  The  general  forms  of  the  third,  fourth, 
etc.,  derivatives  may  be  found  in  like  manner. 

If  in  (1)  and  (2)  x  is  equicrescent,  d?x  —  d?x  =  0,  and  we  have 

^,    and    ?y,  (3) 

while  if  y  is  equicrescent,  d-y  =  d^y  =0,  and  we  have 

dyd^x  3((^xydy —  d^xdydx  ,.. 


dx'  dxP 


58  THE   DIFFERENTIAL   CALCULUS. 

Thus  the  forms  of  the  successive  derivatives,  after  the  first, 
differ,  according  as  the  variable,  the  fvmction,  or  neither,  is 
considered  equicrescent. 

To  transform  a  differential  expression  which  has  been  formed 
on  the  hypothesis  that  x  is  equicrescent  into  its  equivalent  in 
which  neither  x  nor  y  is  equicrescent,  we  have  only  to  re- 
place the  successive  derivatives  by  the  general  forms  (1), 
(2),  etc. 

To  change  the  equicrescent  variable  from  x  to  y,  we  replace 
the  successive  derivatives  by  (4)  directly,  or  by  the  general 
forms,  and  then  make  dry  =  cfy  =  etc.  =  0. 

To  transform  a  differential  expression  formed  on  the  hypoth- 
esis that  either  a;  or  y  is  equicrescent  into  its  equivalent  in  terms 
of  a  new  equicrescent  variable  9,  we  first  replace  the  successive 
derivatives  by  their  general  forms  when  neither  x  nor  y  is  equi- 
crescent, and  then  substitute  for  x,  y,  dy,  dx,  d^y,  d\  etc.,  their 
values  in  terms  of  6. 

Examples.     1.  Change  the  equicrescent  variable  from  xio  y 

in  the  expression  y—  +  —+1  =  0. 
dx^     dxr 

-r,     T     •      d"y  1  dyd^x         ,  dyd^x  ,  dy^  ,  -,      ^ 

Replacing  -^  by  -  -^~-,  we  have  -  y  ^^  +-^-  +  1  =  0, 
dxr  dor  aor        dx- 

or,  dividing  by  dif  and  multiplying  by  da^, 

d^x  _  dx^  _  ^^  _  0 
dy^     dy^      dy 

in  which  the  position  of  dy  indicates  that  y  is  the  equicrescent 
variable. 

2.  Change  the  equicrescent  variable  from  x  to  z  in  the  equa- 

d^v  ■  1 

tion  x*-^  +  a^y  =  0,  having  given  x  =  -- 

dx^  z 

Replacing  ^„  by  ^^^^  -  ^J/^'^  ^e  have,  after  substituting 
^         ^  dx?    ^  daf         '  ' 

J  dz       ^    ,o        2  dz^    dhi  ,  2dy  ,     «       ^ 

dx= and  d^x  = -,    — ^  -\ ^-  +  a-y  =  0. 

z^  ^       dz-      zdz 


SUCCESSIVE   DIFFERENTIATION.  59 

3.  Change  the  equicrescent  variable  from  a;  to  ^  in  the  equa- 
tion — ^ -\ ^  +  ?/  =  0,  having  given  x^  =  4t. 

dor     X  dx 

From   x^  =  4it,   dx  =  — -_,   d^^  = :•     Hence,   replacing 

d^v'  ^^  •  ^  ^^^       d'v      dv 

— ^  by  the  general  form  as  in  Ex.  2,  we  find  t^  -i-  -^  +  w  =  0. 

dx"    ^         ^  '  de      dt      ^ 

4.  Change  the  equicrescent  variable  from  a;  to  ^  in 

d?y         X      dy  ^       ^      =0 
dx^      1  —  x^dx      1  —  X- 

having  given  x  =  sin  6. 

dx  =  cos  6d9,   d^x  =  —  sin  6d6^,    1  —  a;-  =  1  —  sin^  $  =  cos^^. 

Hence 

dxd^y  —  dyd^x  _      x     dy         y      _  cos  6d$d^y  4-  sin  6d£^dy 
dx?  1  —  XT  dx      1  —  a^  cos'^  dd6^ 

sin  9      dy      ,      y         ^         d^y  ,  ^ 

^ —  =  0,  or  — ^  +  «  =  0. 

cos^'e  cos  Odd      cos'' 6  dO'     ^ 

5.  Change  the  equicrescent  variable  from  x  to  6  in  the  ex- 

pression  — < — ^^ ,  having  given  x  =  a  cos  6,  y  —  h  sin  6. 

d^  ^^^    (a^sin^O  +  b^cos^O)'^ 

ah 

C.  If   (a^  —  xr)  — 2=0,  show  that  x'  — -  —  2  =  0, 

dxr      X  dx  dy- 

having  given  x-  +  2/^  =  a". 

We  have  from  x-  +  y^  =  a-,    cZa;  =  —  •"  cZy,   (Z^a;  =  —  - — —• 

X  ar 

Replacing  —  by  ^^^^  ~  ^^^^^'^,  substituting  the  above  values 
dxr  dx' 

of  dx  and  cZ^a;,  and  for  a-—x^  its  equal  ?/-,  the  given  expression 

becomes  a; z  =  0. 

cZ/ 


60  THE   DIFFERENTIAL    CALCULUS. 

7.  Change  the  equicrescent  variable  from  «  to  ^  in  the  ex- 
pression 


dry 
dx- 
having  given  y  =  r  sin  0,  x  =  r  cos  6. 

dy  =  sin  6dr  +  r  cos  6d6,    dx  =  cos  Odr  —  r  sin  9d6. 
(Jpy  =  sin  6drr  +  2  cos  OdBdr  —  r  sin  OdO-. 
d-x  =  cos  ^d-)-  —  2  sin  ^cZ^cZr  —  r  cos  6d^l 
Substituting  these  values,  we  find 


d-t/da;  —  d-a;cZ?/      d'yda;  —  d^xdy      « fZ^'^  _  . d^r 


APPLICATIONS   OP   SUCCESSIVE   DIFFERENTIATION. 

Accelerations. 

dh 
59.  Acceleration.     Signification  of  —  •     Velocity  has  been 

defined  (Art.  6)  as  the  rate  of  change  of  the  distance  passed 

over  by  a  moving  point ;  hence  if  s  be  the  distance  and  v  the 

velocity, 

ds 

''  =  dt' 

The  rate  of  change  of  v  is  called  the  acceleration. 
Now  the  rate  of  change  of  v  is 

fds\ 
dv       \  dt  J      d^s 


dt  dt         df  ' 

fjpg 
hence  —  measures  the  acceleration  of  the  point  in  its  path. 
df 

Being  the  rate  of  v,  the  acceleration  is  the  amount  by  tchich  the 


APPLICATIONS   OF   SUCCESSIVE   DIFFERENTIATION.      61 

velocity  woxdd  change  in  a  unit  of  time  were  the  velocity-rate  to 
become  constant  at  the  instant  considered.  Thus,  if  at  any  in- 
stant the  acceleration  is  said  to  be  5,  it  is  meant  that  at  that 
instant  the  velocity  is  changing  at  the  rate  of  (5  feet  per  sec-  ■ 
end)  per  second,  or  (5  miles  per  hour)  per  hour,  according  as 
the  second  or  hour  is  the  unit  of  time,  and  the  foot  or  mile  the 
unit  of  distance. 

Cor.  1.     If  -y  =  -TT  is  constant,  -tto  =  0,  or  in  uniform  mo- 
dt  dt- 

tion  there  is  no  acceleration. 

doe   dn 
CoR.  2.     Since  — ,  -j-,  are  the  velocities  in  the  directions  of  ^ 

d?x  d^y 
the  axes,  -^,  -~,  are  the  corresponding  accelerations. 

60.    Signs  of  the  axial  accelerations. 

—  may  be  plus  or  minus,  and  the  sign  is  interpreted  as  fol- 
lows :  When  plus,  the  velocity  —  is  accelerated  in  the  positive 

dx 
direction  of  X.     Thus,  suppose  —  is  negative,  or  the  point 

^^  d-x 

moving  in  the  negative  direction  of  X;  then  if  — ^  is  positive, 

the  velocity  is  being  accelerated  in  the  direction  -|-  X,  that  is, 
it  is  algebraically  increasing,  although  numerically  diminish- 
ing, till  the  motion  is  reversed,  after  which  it  increases  numeri- 
cally.    In  other  words,  the  ±  signs  of  the  accelerations  — -, 

j^,  must  be  interpreted  as  an  algebraic  increase  or  decrease  of 

the  corresponding  velocity  whether  the  latter  be  positive  or 
negative. 

Examples.  1.  A  point  moves  in  the  arc  of  the  parabola 
y^=  2px  with  a  constant  velocity  m.  Find  the  accelerations 
in  the  directions  of  the  axis. 

From  the  equation  of  the  path  we  have 

dy^2ldx^  ^-j^^ 

dt      y  dt' 


62  THE  DIFFERENTIAL   CALCULUS. 

and,  by  condition, 

dy 
Substituting  in  this  the  value  of  -~  from  (1),  we  have 


which  in  (1)  gives 


V^      j)^  ^     .    dx  _      my  "  ,f,, 

fdt'  •''  di  ~  V^^+7' 


dy  _      mp 
dt      Vj^Tp 


(3) 


Differentiating  (2)  and  (3),  and  dividing  by  dt  to  obtain 
their  rates,  we  have 

cPx  _      mp''       dy  _      m^p^ 

dry  _  _      mpy      dy  _         m^p^y 
df-~      {y^^p2)^dt~      {y'-\-py 

d-x  . 
Since  —  is  always  positive,  the  velocity  along  X  is  always 

•  d^v 

increasing  algebraically.     — ^  is  negative  in  the  first  angle  and 

positive  in  the  fourth,  hence  the  velocity  along  Y  is  decreasing 
algebraically  in  the  first  angle  and  increasing  algebraically  in 
the  fourth.  These  remarks  are  true  when  the  point  describes 
the  arc  of  the  parabola  in  either  direction. 

d^X        Wi^      d'V  TTV^ 

At  y  =  p,  — ;,=-;—,  — ^  =  —  —  ,  or  at  the  extremity  of  the 
dr     Ap     dr         4p 

focal  ordinate  the  velocities  are  changing  at  the  same  rate. 

2.  A  point  moves  in  the  arc  of  a  circle,  its  horizontal  veloc- 
ity being  9.  Find  the  accelerations  in  the  path  and  along  Y 
at  the  point  x  =  3,  the  radius  of  the  circle  being  5.     From 

ar2_L  7/2— 7?2     dy__xdx__9x__       dx 


dt  y  dt  y  ^E^-x" 


APPLICATIONS   OF   SUCCESSIVE  DIFFERENTIATION.      63 
since  by  condition  —  =  9.     Differentiating  and  dividing  by  dt, 

^  =  0      t^'y  ^     81 J?^  ^  81 R'  , 

df       '      df         'f  (72^-0^)1'  ^^ 

or  the  acceleration  along  X  is  zero,  as  it  should  be  since  the 
motion  in  this  direction  is  uniform,  and  that  along  Y  is  de- 
creasing or  increasing  algebraically  as  y  is  positive  or  negative. 
To  find  the  acceleration  in  the  path,  we  have 


dt       \[(ltj      [dtj       \        y'^dt        y' 


whence       d^s  ^  _9Rdy  ^SlRx^  _S1R^^ 

dt-  y-  dt         f         (i2-'_a^)i 

Making  a;  =  3,  i2  =  5,  in  (1)  and  (2), 

^  =  -31.+     ^=19.+ 
dt^  df 

3.    A  point  moves  in  the  arc  of  a  parabola,  the  velocity  in 

(ilT  ft    S 

the  direction  of  Y  being  constant.     Find  — ,  and  — 

dt  dt^ 

dy  dx     my     ds     m    /-o—, — a   d-s  m^y 

-s.  =  rn,     —  =  — ^,     —  =  —  Vi?  -hy ,  — 5  = ■■' 

dt  dt       p      dt      p  dv     p^p^^y^ 


The  Development  of  Continuous  Functions. 

61.  Limit  of  a  variable.  The  limit  of  a  variable  is  that  value 
which  it  constantly  ap])roaches  hut  never  reaches. 

Thus,  the  limit  of  a;  =  1  +  |  +  ^-f  i-H is  2. 

The  statement  that  2  is  the  limit  of  x  implies  a  particular  law  of  in- 
crease. If  X  increases  by  the  successive  additions  of  |^  to  1,  2  is  not  the 
limit  of  x  =  1  +  J  +  J  +  •••,  for  by  the  law  of  its  increase  x  can  be  made  to 
exceed  2  in  value.  But  x=\-\-^-\-\-\-\-\----  can  never  become  equal  to 
2,  since  by  the  law  of  its  change  each  increment  is  but  half  the  difference 
between  2  and  the  value  of  x  at  any  instant.  So  if  a  circle  be  circum- 
scribed about  a  regular  polygon,  its  area  is  not  the  limit  of  the  area  of  the 


64  THE   DIFFERENTIAL   CALCULUS. 

polygon  if  the  polygon  changes  by  the  motion  of  its  vertices  along  the  pro- 
duced radii ;  for  in  that  case  the  area  of  the  polygon  may  become  greater 
than  that  of  the  circle.  But  if  the  number  of  sides  of  an  inscribed  polygon 
be  indefinitely  increased,  its  vertices  remaining  in  the  circle,  the  area  of 
the  circle  is  the  limit  of  that  of  the  polygon,  since  no  inscribed  polygon,' 
however  many  its  sides,  can  coincide  with  the  circle.  I 

It  is  evident  that  if  we  conceive  the  law  of  change  of  a  vari- 
able to  continue  indefinitely  in  operation,  the  variable  may  be 
made  to  approach  as  nearly  as  we  please  to  its  limit.  Hence 
the  difference  between  a  variable  and  its  limit  is  itself  a  variable 
whose  limit  is  zero. 

62.  The  term  limit  is  also  applied  to  a  magnitude  of  varying 
position  as  well  as  to  one  of  varying  value.  Thus,  OT,  the 
tangent  to  MN  at  0,  is  said  to  be  the  limit 

of  the  secant  OP,  since  the  secant,  having  at 

least  two  points  in  common  with  the  curve 

by  definition,   can  never  coincide  with  the 

tangent ;  or,  more  properly,  6  is  the  limit  of 

^  as  P  approaches  0.     Observe  that,  as  in 

the  previous  illustrations,  if  P  approaches  0 

without  condition,  $  is  not  the  limit  of  ^; 

but  if  we  affix  the  condition  '  OP  remaining 

a  secant,'  then  6  is  the  limit  of  <^,  P  being  made  to  approach 

as  near  as  we  please  to  0  but  not  coinciding  with  it. 

63.  The  term  limit  is  frequently  used  with  another  meaning 
which  must  be  carefully  distinguished  from  that  above  ex- 
plained. Thus  ±  R  are  said  to  be  the  limiting  values  of  x 
and  y  in  the  equation  a;-  +  y^  =  Pr.  To  distinguish  such  limit- 
ing values  of  a  variable  from  one  which  the  variable  approaches 
but  never  reaches,  the  latter  is  often  written  x  =  2,  ^  =  6, 
which  in  the  illustrations  of  Arts.  61  and  62  are  read  '  x  ap- 
proaches 2  as  a  limit,'  as  the  number  of  terms  of  the  series 
increases  indefinitely,  '^  approaches  ^  as  a  limit,'  as  P  ap- 
proaches 0  (Fig.  10). 


APPLICATIONS   OF   SUCCESSIVE   DIFFERENTIATION.      65 

64.  It  is  evident  from  the  definition  that  a  quantity  cannot 
approach  two  limits  simultaneously.     Thus,  if  2  is  the  limit  of 

a;  =  l+:^  +  ^-j ,  X  can  be  made  to  approach  2  in  value  as 

near  as  we  please,  and  therefore  no  value  less  than  2  can  be  its 
limit;  nor  can  any  value  greater  than  2  be  its  limit,  since  it 
can  never  equal  2  and  therefore  cannot  be  made  to  approach 
any  value  greater  than  2  as  near  as  we  please. 

65.  Continuous  functions.  A  function  of  a  variable  is  con- 
tinuous between  certain  values  of  the  variable  when  it  has  a 
finite  value  for  every  intermediate  value  of  the  variable  and 
changes  gradually  as  the  variable  so  changes  from  one  value  to 
the  other. 

Thus,  in  y  =  mx  +  6,  y  is  a  continuous  function  of  x  for  all 
values  of  x ;  in  iC-y^  +  h-'j?  =  arb^,  y  is  continuous  between 
x  =  ±a;  in  aV  —  ^^^  =  —  <*"^^  V  is  discontinuous  between 
X  =  ±  a,  and  continuous  for  values  of  aj  >  a  numerically  ;  in 
xy  =  m,  y  is  discontinuous  for  x  =  0.  And,  in  general,  if  ?/  is  a 
continuous  function  for  all  values  of  x,  y—f{x)  represents  a 
curve  of  unbroken  extent.  _^ 

m  66.  Series.  A  succession  qj  terms  ivhicfi  follow  each  other 
iticcording  to  some  law  is  called  a  series.  When  known,  the  law 
enables  us  to  determine  any  term  of  the  series. 

A  series  is  finite  or  infinite  as  the  number  of  its  terms  is 
limited  or  unlimited. 

67.  The  sum  of  a  finite  series  is  the  sum  of  its  terms. 

The  sum  of  an  infinite  series  is  that  finite  limit  whose  value  the 
sum  of  its  terms  continually  approaches  as  the  number  of  terms 
increases.  If  there  be  no  such  finite  value,  the  series  is  diver- 
gent ;  if  such  a  value  exists,  the  series  is  convergent. 

68.  To  develop  a  function  is  to  find  a  series  whose  sum  is 
equal  to  the  function.  The  development  of  a  function  is  there- 
fore a  finite  or  an  infinite  converging  series ;  in  the  former  case 


66  THE   DIFFERENTIAL   CALCULUS. 

the  function  being  the  sum  of  the  terms,  and  in  the  latter  the 
limit  of  the  sura  of  the  terms. 

When  the  series  is  converging,  the  difference  between  the 
function  and  the  sum  of  the  first  n  +  1  terms  of  the  series  is 
called  the  remainder  after  n  + 1  terms,  and  the  limit  of  this 
remainder  as  n  increases  must  evidently  be  zero. 

Illustrations.  A  function  may  be  developed  by  involution  when  its 
exponent  is  a  positive  integer.  Thus  (1  +  x)^=  1  +  3  a;  +  3 x^  +  «^,  a  finite 
series,  whose  sum  is  equal  to  the  fimction,  and  which  is  therefore  its 
development. 

A  function  may  be  developed  by  division  if  the  indicated  division  can 

X*  — 1 

be  completed.    Thus,  =  x'  +  x+  1,  a  finite  series.    When  the  divisor 

X  — 1 

is  not  exactly  contained  in  the  dividend,  division  leads  to  an  infinite  series, 

as  =l  +  x4-x2-fx^H — ,  and  the  process  also  furnishes  the  remain- 

1— X 
der  after  n  +  1  terms.      Since  this  remainder,  when  added  to  the  tenns 
already  found,  must  equal  the  function,  it  must  decrease  as  n  increases, 
and  its  examination  will  discover  whether  the  series  is  or  is  not  converg- 
ing, that  is,  whether  it  is  or  is  not  the  developaient  of  the  function.   Thus, 

in  the  above  case,  the  remainder  after  n  +  1  terms  is  - — ,  which  decreases 

1-x 
as  n  increases,  only  when  x  <  1.     Hence  if  x  <  1,  the  series  is  converging, 

and  we  may  virrite   =  1+ x  +  x^+x^-l- •••,  understanding  that  the 

1  — X 
second  number  approximates  more  closely  in  value  to  the  first  as  the  series 
is  extended;  while  if  x  >  1,  the  series  is  diverging,  and  cannot  be  equal  to 
the  function,  or  is  not  its  development. 

Other  processes  of  deriving  a  series  from  a  function  do  not  afford  the 
remainder,  and  tlnis  do  not  indicate  whether  the  series  diverges  or  con- 
verges. Thus  evolution,  or  the  extraction  of  the  root  of  a  polynomial, 
fiu-nishes  in  general  an  infinite  series,  but  no  remainder. 

No  imiversal  criterion  for  determining  whether  a  given  series  is  converg- 
ing or  diverging  has  been  found. 

69.  Maclaurin's  theorem.  The  object  of  Maclaunn's  theorem 
is  the  development  of  a  function  of  a  single  variable  into  a  series 
arranged  according  to  the  ascending  powers  of  the  variable  with 
finite  and  constant  coefficients. 

The  proposed  development  will  be  of  the  form 

f(x)  =A  +  Bx  +  Cx'  +  X>.^-'  +  Ex*+--;  ( 1 ) 


APPLICATIONS   OF   SUCCESSIVE  DIFFERENTIATION.      67 

in  which  A,  B,  C,  etc.,  are  finite  and  independent  of  x.  It  is 
required  to  find  such  values  for  A,  B,  C,  etc.,  as  will  satisfy  (1) 
for  all  values  of  x,  that  is,  render  the  series  either  finite,  or,  if 
infinite,  then  converging. 

Since  (1)  is  to  be  true  for  all  values  of  x,  it  must  be  true  for 
ic  =  0  ;  whence  A  =f(x)  when  a;  =  0,  or  the  first  term  of  the 
series  is  what  the  function  becomes  when  x  =  0.  Differentiat- 
ing (1),  the  successive  derivatives  are 

/'  (a;)  =B  +  2Cx-\-3  Dx"  +  4  Ex-  -f  •  •  -, 

/"  (X-)  =  2  C  +  2  .  3  Z).«  +  3  •  4  ^^2  +  ..., 

/'"(a;)  =  2  .  3  i)  +  2  .  3  •  4  ^o;  +  •  •  •, 
etc., 
which,  being  true  for  all  values  of  x,  are  true  for  a;  =  0.    Hence 
representing  by /(()),/' (0),/"(0),  etc.,  what/(a;),/'(a)),/"(a;), 
etc.,  become  when  x—0,  we  have 

i^=/'(0), 
2C=/"(0),     .-.  0  =  -^"^^^ 

2-3Z)=/"'(0),    .-.  D  = 
etc., 
and  substituting  these  values  in  (1), 

f{x)  =/(0)  +/'(0)a:  +/"(0)  |  +/"'(0)  t  +  ...,        (2) 

and  the  theorem  may  be  thus  stated : 

The  first  term  of  the  series  is  lohat  the  function  becomes  when 
a;  =  0  ;  the  second  term  is  what  the  first  derivative  of  the  function 
becomes  when  a;  =  0,  into  x;  the  third  term  is  what  the  second 
derivative  of  the  function  becomes  ivhen  x  =  0,  into  x'  divided  by 
factorial  2;  and,  in  general,  the  {n  +  \)th  term  is  what  the  nth 
derivative  of  the  function  becomes  when  aj  =  0,  into  aj"  divided  by 
factorial  n. 


68  THE   DIFFERENTIAL   CALCULUS. 

If  the  resulting  series  is  finite,  it  is  equal  to  the  function, 
the  two  members  of  (2)  are  identical,  and  the  development  is 
effected.  If  the  resulting  series  is  infinite,  it  is  necessary  to 
determine  whether  it  is  convergent. 

70.  Taylor's  theorem.  The  object  of  Taylor's  theorem  is  the 
development  of  a  f  auction  of  the  algebraic  sum  of  two  variables 
into  a  series  arranged  according  to  the  ascending  powers  of  one  of 
the  variables,  with  finite  coefficients  depending  upon  the  other  and 
the  constants  which  enter  the  function. 

The  proposed  development  will  be  of  the  form 

f{x  +  y)  =  P+  Qy  +  Rf  +  Sf  +  :.,  (1) 

in  which  P,  Q,  R,  etc.,  are  functions  of  x,  and  independent  of  y. 
It  is  required  to  find  such  values  of  P,  Q,  R,  etc.,  as  will  satisfy 
(1)  for  all  values  of  x  and  y,  that  is,  render  the  series  finite, 
or,  if  infinite,  then  converging. 

Since  (1)  is  to  be  true  for  all  values  of  x  and  y,  it  must  be 
true  when  y=0;  in  which  case  P=f{x),  or  the  first  term  of 
the  series  is  what  the  function  becomes  when  2/  =  0. 

Let  a  be  any  value  of  x,  and  P',  Q',  R',  etc.,  the  correspond- 
ing values  of  the  coefficients,  which  are  functions  of  x.  Then 
(1)  is  true  for  x  =  a,  and  we  have 

/(a  +  y)  =  P'  +  Q'y  +  R'-f  +  S'f  +  •  •  •,  (2) 

whose  successive  derivatives  are 

/'  (a  +  2/)  =  Q'  +  2P'.v  +  3.Sy..., 

f'{a  +  y)=2R'  +  2-'SS'y:., 

/'"(a +  2/).  =  2.  3^'..., 

etc. 

Since  these  equations  must  be  true  for  all  values  of  y,  they 
are  true  for  y  =  0.     Hence 

f"(a)  f"'(a) 

p'=f{a),   Q'=f\a),   i^'  =  •^^   ^'  =  -^'  ^*^- 


APPLICATIONS   OF   SUCCESSIVE   DIFFERENTIATION.      69 

Substituting  these  values  of  P',  Q',  R',  etc.,  in  (2), 

/(a  +  y)  =f{a)  +f>(a)y+f"{a)  t  +/"'(a)  |  ••., 

in  which  the  coefficients  are  what  f{x),  f'{x),  f"{x),  etc.,  be- 
come when  x  =  a.  But  a  is  any  arbitrary  value ;  hence,  what- 
ever the  value  of  x, 

f{x  +  y)=f(x)+f'{x)y+f"{x)y^  +  f"'{x)t...^ 

and  the  theorem  may  be  thus  stated : 

The  first  term  of  the  series  is  what  the  function  becomes  when 
y=Q',  the  second  term  is  the  first  derivative  of  the  function  when 
2/  =  0,  into  y ;  the  third  term  is  the  second  derivative  of  the  func- 
tion ichen  2/  =  0,  into  y^  divided  by  factorial  2  ;  and,  in  general, 
the  (n-}-l)th  term  is  the  nth  derivative  of  the  function  when 
y  =  0,  into  t/"  divided  by  factorial  n. 

As  before,  if  the  series  thus  obtained  is  infinite,  it  is  neces- 
sary to  determine  whether  it  is  convergent. 

71i  Completion  of  Taylor's  and  Maclanrin's  Formolae. 

Since  the  use  of  infinite  eeriea  as  the  equivalents  of  the  functions  is  inadmissible 
unless  the  series  are  converging,  it  is  necessary  to  determine  the  remainder  after  m  + 1 
terms  in  the  preceding  formula;,  and  to  examine  this  remainder  in  any  particular  case  to 
see  if  its  limit  is  zero  as  n  increases. 

I.  Iff{.x)  becomes  zero  when  x=a  and  x  =  h,  and  is  continuous  between  these  val- 
ues, and  iff'{x)  is  also  continuous  between  these  values,  then  f'{x)  will  be  zero  for 
some  value  ofx  between  a  and  b. 

For,  since  /{x)  =  0  for  x=  a  and  x  =b,  as  x  changes  from  a  to  b,  /(a;)  must  either 
first  increase  and  then  decrease,  or  first  decrease  and  then  increase.  But  the  iirst  deriva- 
tive is  positive  when  the  function  is  increasing  and  negative  wlien  it  is  decreasing  (Art. 
22),  and  therefore  in  either  case  it  changes  sign  between  the  values  x=a  and  x=b;  and 
being  continuous,  it  cannot  become  infinite,  and  therefore  must  pass  through  zero. 

II.  First  form  of  the  remainder. 
Resuming  Taylor's  formula, 

a)Ax  +  y)=Ax)+f(.x)y+f(x)y^+J-(x)'^...+f'>(x)'^+": 

Writing  x  +  y=  X,  whence  y=  X -x,  and  representing  by  B  the  remainder  after 
n  + 1  terms,  we  have 

(2)  /(X)  =/(«)  +f(x)(X-x)  +f(x)  ^^~^^'+/"W  ^^~^^'- 


70  THE   DIFFERENTIAL   CALCULUS. 

(A'  — a;)"+^ 
Writing  the  remainder  in  the  form  P-^^. ^ — »  P  being  a  function  of  X  and  x  ti 

be  determined,  substituting  this  value  of  R,  and  transposing,. we  obtain 

(3)  AX)  -f{x)  -f(x)iX-x)  -fix)  H^' _/■'(«)  ^^§^'- 

•'    ^  ^       \_n  \n  +  l 

Representing  by  F{z)  the  function  of  z  which  (3)  becomes  by  substituting  z  for  x, 

(X—z)^  (X—z)3 

(4)  Fiz)=/{X)  -/(«)  -f{z){X-z)  -/"(g)^    |/    -/"'(g)  ^    ^       ••• 


I  »  I  w  +  1 

If  2=  a:  in  (4),  it  becomes  identical  with  (3)  and  therefore  =  0.  It  also  becomes  zero 
if  z=  X,  for  every  term  then  contains  a  zero  factor.  Therefore,  by  I.,  its  derivative 
F'(,z)  must  be  zero  for  some  value  of  z  between  x  and  A'.  If  S  be  a  proper  fraction, 
z=  x-i-  9(X  —  x)  will  represent  such  iutermcdiate  value. 

Differentiating  (4)  to  obtain  F'{z),  we  have 

F\z)  =  0-f{z)+f(,z)-f(z)(X~z)+f{z)iX-z) 

-f(Z)'-^+f(z)'—^-.:+riZ)'-^^-^ 

!_n  (_M 

whose  terms  vanish  in  pairs,  except  the  last  two,  giving 

^,.,._,«,„«^.,<^. 

Substituting  the  value  z  =  x  +  d(_X—x)  for  which  F' (z)  is  zero,  we  have,  after  can- 

(X—z)^ 
celline  the  common  factor  , 

[n- 

or  -f'^+^lx  +  e{X-x)]  +  P=0, 

y    p=/n+i[x  +  e(A'-a;)], 

in  which  all  we  know  of  0  is  that  its  value  lies  between  0  and  1. 
Hence  the  remainder  after  n  +  1  terms  is 

I «  +1* 


/     Ji-P  ^'^;f r'  =/"+^[^  +  KX-X)^  (^^Z|)^1'^/n+l(^  +  gy)^j^ 


(5)  fix  +  y)=fix)+fix)y  +fix)  y^...  +/»(a;)  ?J+/»+i(a;+9y)  ^. 


and,  substituting  in  (1),  the  completed  form  of  Taylor's  theorem  ia 

Making  a;  =  0,  and  changing  y  to  x, 

(6)  /(a;)  =/(0)+/(0)a;  +/"(0)  ^' ...  +/»(0)  g  +/»+l(to)  j^. 

the  completed  form  of  Maclaurin's  theorem. 

"We  thus  have  in  both  cases  the  remainder  after  n  +  \  terms,  which  is  found  by  dif. 


APPLICATIONS    OF   SUCCESSIVE   DIFFERENTIATION.      71 

ferentiating  the  function  »t  +  1  times  and  changing  a;  to  a;  +  9y,  or  to  0x,  in  the  (n  +  l)th 
derivative,  and  multiplying  this  result  by   "^ j,  or  by  , — --^-    If  this  remainder  is 

zero,  the  series  is  finite  ,•  if  its  limit  is  zero  as  n  increases,  the  series  is  convergent: 
if  it  is  neither  zero  nor  lias  zero  for  a  limit,  the  formulas  fail. 

III.  Second  form  of  the  remainder. 

Writing  the  remainder  in  the  form  It  =  I\(X-  x),  (3)  would  become 

(7)  /(A-)  ~Ax)  -f{x)  (-Y-  x)-f  '{x)  ^-^^^'  -  -f'\x)  ^^^^f^-  A(-V-  X)  =  0. 

Kex>reBenting  by  F{z)  what  (7)  becomes  by  substituting  z  for  x, 

F(z)=f(X)-f(z)-f(z)(X-  z)-f"{z)  —^  --/"(s)  ^^^^  -  AC-V-s), 

in  which,  if  a  =  a;  or  «=T,  jPCs)  =  0  as  before ;  and  therefore  F  {z)  =  (iiov  z=  x  + 6(^X—x). 
Differentiating  to  find  F'{z),  the  terms  vanish  in  pairs  except  the  last  two,  giving 

F'(.z)  =  -r'+\z)^'  ^'  +p„ 

and,  substituting  the  value  of  z  for  which  F\z)  =  0, 

V                                                                                            ..n+lr-i  _  g\n 
and  R=  Pi(X- X)  =f"+^(x  +  Oy)  ^ ^ —• 

Substituting  in  (1),  a  second  completed  form  of  Taylor's  formula  is 

(8)  A^  +  y)  =/(«)  +fWy  +/•  C^) |f  -  +/"(^) ^  +/"+'(a;  +  <?y)  ^""*"  ^(^~*^"- 
Making  x  =  0  and  changing  ^  to  a;,  the  corresponding  form  of  Maclaurin's  formula  is 

(9)  Ax)  =/X0)  +/-(0)a;+/'(0)g...+/»(0)g+/»+l(9x)^""^  [L'"^"' 
to  which  forms  apply  the  remarks  made  upon  (5)  and  (6) . 

IV.  If  the  (n  +  l)th  derivative  is  finite  for  all  values  of  n,  Taylor's  and  Maclati- 
rin'sformulte  develop  f{x  +  y)  andf{x),  respectively. 

The  first  forms  of  the  remainder  are 

R=f^+\x  +  9y)''~  and  R  =  f^^^  ^ex) '^. 
But  when  n  + 1,  as  n  increases,  becomes  equal  to  x, begins  and  continues  to 

|Tt  +  l 

diminish,  each  successive  value  being  less  than  the  preceding  one.    Hence,  whatever 

_    n+l 

the  value  of  x,  provided  only  it  be  finite,  as  it  is  by  hypothesis, tends  to  the  limit 

\n  +  \ 

zero  as  n  increases  indefinitely.    It  follows,  therefore,  that  if  the  (»t  + 1)  th  derivative 


72 


THE   DIFFERENTIAL   CALCULUS. 


does  not  become  influite  with  n,  R  approaches  zero  as  n  increases,  and  the  serieB  is 
convergent. 

The  same  is  also  true  of  the  second  forms  of  R. 

V.  It  is  evident  that  the  sum  of  the  first  n  terms  of  a  series  cannot  approach  a  fixed 
value  as  n  increases  indefinitely,  unless  the  terras  finally  decrease;  that  is,  unless  the 
ratio  of  the  nth  term  to  the  one  before  it  becomes  and  continues  less  than  unity  as  n 
increases,  the  series  cannot  be  convergent. 

72.  Applications.  Assuming  that  the  following  functions 
can  be  developed,  show  that : 

1.    (a  +  xy  =  aJ  + 7  a^x  + 21  aV  -f  35  aV  +  35  aV  -\-  21  aV 

+  7  aa^  -\-  x'  ' 
Making  a;  =  0,  /(O)  =  a'. 
Tlie  successive  derivatives  are : 


/'   (x)  =  7 (a  +  a;)",  whence  /'  (0)  =  7 a*' 

f"  {x)  =  G-7{a  +  xy,  "  /"(0)  =  6 

f"'{x)  =  5'6'7{a  +  xy,  «  /"'(0)  =  5 

/'  (a;)  =  4  .  5  .  6  •  7(a  +  xf,  "  f  (0)  =  4 

r  (x)  =  3.4.5.C.7(a  +  a;)S     "  ^  (0)  =  3 

/'^  (a;)  =  2  .  3  •  4  .  5  .  6  •  7(a  +  a;),  "  /"  (0)  =  2 

p'"(x)  =  2.3.4.5.6.7,  "  /'"(O)  =  2 

r'"Xx)  =  0. 

Substituting  in  Maclaurin's  formula, 

/(a.)=/(0)+/'(0)x+/"(0)^'  +  /"'(0)^..., 

we  have 

(a  +  x)^  =  a^  +  7a''a:  +  6  .  7a^  ^  +  5  .  6  .  7a^^  +  4  •  5  .  6  .  7a'' 


■  (S'7a\ 

•  ^•Q>-7a\ 

•  4.5.6.7a2. 
.3.4.5.6.7a. 
.3.4.5.6.7. 


[2 


[3 


li 


+  3. 4.5.6.  7a2 -  +  2.3. 4. 5. 6.7a- 

[5  [6 


+  2.3.4.5-6.7 


iZ 


=  a'  +  7  a^x  +  21  a^a;^  -j,  35  a'x^  +  35  aV  +  21  aV 
+  7aa;''  +  a;^ 


APPLICATIONS    OF   SUCCESSIVE   DIFFERENTIATION.      73 

Being  finite,  the  series  is  the  development  of  the  function, 
as  will  evidently  be  the  case  so  long  as  the  exponent  of  the 
binomial  is  a  positive  integer. 

rvtS  /y^  rv*t 

2.  sm. =  .--  +  ---. .. 

Making  a;  =  0,  /(O)  =  0. 

The  successive  derivatives  are . 

/'  (cc)  =  cos  X,     whence  /'  (0)  =  1, 
/"  (rr)  =  -  sin  x,       "       /"  (0)  =  0. 
/'"(a;)  =  -  cos  X,      «       /'"(O)  =  -  1. 
f"  {x)  =  sin  X,  "       f\^)  =  0. 

Since  f"{x)  is  the  original  function,  these  values  will  recur 

in  sets  of  four,  and  we  have 

a^  ,  0^     x^ 
since  =  x .. 

[3     |5      \l 

3                                                     0*^            '>•'*            "7*" 
.  cosa;=l , — . 

[2     li     L5 

Since  the  (n  +  l)th  derivatives  of  sinx  and  cos  a;  are  finite  whatever  the  value  of  n, 
the  formula  develops  these  functions  (Art.  71,  FV.),  and  the  error  may  be  made  as  small 
as  we  please  by  taking  a  sufScient  number  of  terms. 

By  means  of  these  series  we  may  compute  the  natural  sine 
or  cosine  of  any  arc,  but  few  terms  being  necessary  as  the 

series   convel-ge   rapidly.      Thus,  if   x  =  —  =  .174533  be  sub- 

18 
stituted  for  x  in  the  series  of  Ex.  2,  sin  x  =  sin  10°  =  .173G5+. 

4.  a'  =  1  +  log  a  ■  x  +  log^  a — \-  log*a  — h  •••• 

[2  [3 

Making  x  =  0,  /(O)  =  a"  =  1. 
The  successive  derivatives  are 

f(x)  =  a'loga,  f"{x)  =  a'\og^a,  f"(x)  =  a'log'a,  etc. ; 
whence 

/'(O)  =  loga,  /"(0)  =  log2a,  /"'(O)  =  log«a,  etc.. 


74  THE  DIFFERENTIAL  CALCULUS. 

which,   substituted   in   Maclaurin's   formula,  give   the   above 
series.     Making  a  =  e,  whence  log  e  =  1,  it  becomes 

/>•-  /yJ  o** 

|2     [3     [4     ' 
and  if  x  =  1, 

e  =  1  +  1 +,^  +  ,4  +  ,4- =2.718281+ 

[2     [3     U 

the  Napierian  base.     These  are  the  exponential  series. 

The  (»i  +  l)th  derivative  of  o^  is  (loga)"+^a*,  and  hence 

•^  ^|W  +  1  jW  +  1 

But  a"-^  is  finite,  and 

(r  log  a)""^  _  a;  log  a    a:  log  a     a;  log  a 
r»  +  l       ~      i  2      '"  n  +  1  ' 

which  approaches  zero  as  Ji  increases ;  therefore  the  formula  develops  a'. 

K    /i  I     \m      1  ,  1  wi(m  — 1)    o,  m(m— l)(m  — 2)  , 


-I- ^(^-l)---(^-^  +  l)a;n  I  .. 


To  determine  for  what  values  of  x  the  formula  develops  (1  +  ar)*". 
The  (n  +  l)th  derivative,  when  9x  is  written  for  x,  is 

m(,m  -  1)  —  (TO  -  n)  (1  +  ea:)"'"''"\ 

which  bec6mes  zero  if  m  is  a  positive  integer  when  n=  in.  Hence  the  series  is  finite,  and 
is  the  development  of  (1  +  x)'"  when  m  is  a  positive  integer.  If  m  is  negative  or  frac- 
tional, the  series  >s  infinite.    The  ratio  of  its  nth  term  to  the  one  immediately  before  it  is 

m  —  n  +  1        /TO  +  1     ,\ 

x  =  \ T-JX, 

n  ^     n  ' 

whose  absolute  value,  as  n  increases,  will  eventually  become  and  remain  greater  than 
unity  if  x  is  numerically  greater  than  1.  Hence  (Art.  71,  V.)  the  series  is  divergent,  and 
cannot  equal  (1  +  a;)"*  when  x  is  numerically  greater  than  1.  The  remainder  after  n  +  1 
terms  is 

b"+^      rTO(?»-l)  •••  (to  — 7!)    „,i-|  1 


R ^fn+H6x)  fZL   rm(TO-  )-(TO-7»)^„^n 

■'  n+1     L  n  +  1  J 


(1  +  6x)" 


When  X  lies  between  0  and  1,  the  last  factor  becomes  less  than  1  as  n  increases.    In- 
creasing n  by  1  multiplies  the  first  factor  by — x,  or  ( ~ ^x,  which 

approaches  —a:  as  n  increases;  that  is,  a  quantity  numerically  less  than  1.  Hence  to 
increase  n  indefinitely  is  to  multiply  by  an  infinite  number  of  factors  each  less  than  1 ; 
the  product  therefore  decreases  indefinitely,  and  the  formula  develops  (1  +  a.')"  for  values 
of  X  between  0  and  1.    By  means  of  the  second  form  of  the  remainder  we  have 


APPLICATIONS    OF   SUCCESSIVE   DIFFERENTIATION.      75 

rOT(OT-l)-(m-n)       ,1-1  /l-9\"+i  (1  +  to)"» 
=  L  [n  JU  +  ea;;  1-9     ' 

When  X  lies  between  0  and  -1,  the  last  factor  is  finite;  (  ^— ^ — )"       approaches  zero 

VH-  ex' 

as  n  increases ;  increasing  n  by  1  multiplies  the  first  factor  by  ^  ~  "  ~    x,  which  ap- 

n  +  \ 
proaches  —  x  as  w  increases.    Hence,  as  before,  the  formula  develops  (1  +  ar)™  for  values 
of  X  between  0  and  —  1. 

Since  (a  +  x)"*  may  be  written  in  either  of  the  forms 

«'»(i.3-,  x".(i.^)» 

and  as  one  of  those  can  be  developed,  whatever  the  relative  values  of  a  and  x,  the 
Binomial  formula  holds  good  for  fractional  and  negative  exponents.  When  m  is  a 
positive  integer,  the  series  is  finite,  and  the  formula  holds  good  for  both  the  above  forms. 

°^  ^  2      3      4 

Making  x  =  0,  /(O)  =  log  1  =  0. 
The  successive  derivatives  are 


1+x             •'    ^  ^          {l  +  xf 
f"(x)  = ? ,       fUx)  = ll^  etc. ; 

whence/'(0)  =  l,/'(0)=-l,/"(0)  =  2,/-(0)  =  -2.3,  etc., 
and  these  in  Maclaurin's  formula  give  the  above  series. 

The  ratio  of  the  nth  term  to  the  preceding  one  is  (~^M"  "  '■'x  or  —  (1  —  l)x,  which, 

n  \       «/ 

if  xis  numerically  greater  than  1,  becomes  and  remains  greater  than  unity  as  n  increases; 
hence  (Art.  71,  V.)  the  series  is  divergent  if  x  is  numerically  greater  than  1.    The  (n  +  l)th 

In 
derivative  is  (—1)"        '—        ,  and,  using  the  first  form  of  R, 

(1  +  x)''+i 

•'        ^       |n  +  l      n  +  1  ^\  +  ex' 

If  X  lies  between  0  and  +  1,  — ^—  is  a  proper  fraction,  and  R  approaches  zero  as  n 
1  +  ex 
increases. 

If  x  lies  between  0  and  —1,  the  series  becomes  ~x  —  - —      ...,  and  the  second  form 

2       3 
of  R  gives,  numerically, 

'      ^    '         [n  \\-ex'  i-ex 


76  THE  DIFFERENTIAL   CALCULUS. 

For  valneB  of  x  between  0  and  1,  /^~   ^y  is  a  proper  fraction,  and  approaches  zero 

as  n  increases,  while  the  last  factor  is  finite.  Ilence  the  formula  develops  log  (1  +  x) 
when  X  lies  between  +  1  and  —1. 

7.   log.(l  +  x)  =  ™(x-|  +  f-|  +  f...);  (1) 

if  a  =  e,  we  have,  as  in  Ex.  6, 

/y*~  /y^  /y*'*  O*^ 

log  (l+x)  =  x- -+---  +  --,  (2) 

^^^  2345  ^  ^ 

which  are  the  logarithmic  series.  As  they  diverge,  if  a;  >  1, 
they  are  not  suitable  for  the  computation  of  logaritlims.  To 
adapt  them  to  this  purpose,  substitute  —  x  for  a;  in  (1),  and 
we  have 

/  rt*2  /yH>  /}*4  ™,5  \ 

log,(l-.)  =  ™(_.-|-|-|-|...).  (3) 

Subtracting  (3)  from  (1), 
log„ (1+a^)- log,.  (l-rr)  =  2m  |a;  +  |  +  |  +  y +  •••}• 

Let  X  = :  then  x  is  less  than  1  for  all  positive  values 

22  +  1'  '■ 

of  z,  and 

log„  (1  +  a)  -  log„  (1  -  x)  =  log„-±^  =  log„^-±- 

1  —  X  z 

=  log„(2;  +  l)-log„2; 

=  2m\-^A ^ + ^ I,    (4) 

or,  if  a  =  e,  whence  m  =  1, 

logCz+l)  -log2  =  2  j  -^—  -\ ^ H ^ \  . 

''^        ^        ''  (22+1     3(22+1)3^5(22+1)^      j 

From  this  series,  which  converges  rapidly,  we  may  compute 
the  Naperian  logarithms  of  numbers.  Thus,  if  2  =  1,  log  1=0, 
and  we  have 

log2  =  2|-  +  -^  +  -^+^-,+  "-  !-  =.693147+, 
^  (3     3-3='     5-3^     7-3^  i  ' 

when  six  terms  are  taken. 


APPLICATIONS   OF   SUCCESSIVE   DIFFEKENTIATION.      77 

Making  z  =  2, 

log3=log2+2|l+^+Jj,  +  ^+...}=1.0986m. 

log  4=2  log  2  =  1.386294+. 

Making  x  =  4, 

log5=log4+2{--f^+-^+-^,+  ---|  =  1.6094379+. 

In  like  manner,  the  Naperian  logarithms  of  all  numbers  may 
be  computed. 

Cor.  1.    The  Naperian  logarithtn  of  the  base  of  the  common 
system  is      ^^^  ^^  =  log  5  +  log  2  =  2.302585+. 

Cob.  2.   From  (4),  b  being  the  base  of  the  system,  and  m' 
the  corresponding  modulus, 

^*    z  12^  +  1      3(2;z  +  l)^  i        ^  ^ 

Since  (4)  and  (5)  are  true  for  all  positive  values  of  z,  writ- 
ing X  for  ,  we  have 


z 

log^x     m 


(6) 


logj  X     m' 

or  the  logarithms  of  the  same  number  in  different  systems  are 
proportional  to  the  moduli  of  the  systems. 

CoR.  3.    If  in  (6)  b  =  e,  then  m'  =  1,  and 

log„  ic  =  m  log  X.  (7) 

Having  then  computed,  as  above,  a  table  of  Naperian  loga- 
rithms, the  logarithms  in  any  system  may  be  found  by  multiply- 
ing their  Naperian  logarithms  by  the  modidus  of  the  system. 

Con.  4.    Since  log„  a  =  1,  if  a;  =  a  in  (7), 

1 

m= , 

log  a 


78  THE  DIFFERENTIAL  CALCULUS. 

or  the  mochihis  of  any  system  is  the  reciprocal  of  the  Naperian 
logarithm  of  its  base ;  which  is  the  relation  between  the  mod- 
ulus of  a  system  and  its  base  referred  to  in  Art.  28. 

CoR.  5.     In  the  common  system  a  =  10,  hence 
1  1 


m 


—  =  .434294+, 


log  10     2.302585 
the  modulus  of  the  common  system. 

rjM'i  nfl^  rv** 

8.  tan~^a;  =  rc 1 . 

3       5       7 

Making  a?  =  0,  /(O)  =  0. 

The  first  derivative  is =  1  —  ic^  -f  a;''  —  «'"'  +  a/*  —  oj""  •  •  • 

by  division ;  hence  the  successive  derivatives  are 
/'  {x)  =  l-x'  +  x^-  x"  +  x^-  a;'"  ••., 
/"  (a;)  =  -  2x'  +  4 ar^  -  Ga:*  +  8a;'  -  10a;»..., 
/'"(a;)  =  -  2  +  3 •  4a^  -  5  •  6a;^  +  7 . 8a*  -  9  •  lOa^ ..., 
/' (a;)  =  2 . 3 .4a;  -  4 . 5 .  Gar* -f  G . 7 . 8ar' -  8 . 9 •  10a;' ..., 
/'  (a;)  =  2.3-4-3.4.5-Ga;^  +  -.-, 

from  which 

/'(0)=1,        /"'(0)  =  -2,        /^(0)  =  2.3.4, 
/"(0)  =  0,        /'^(0)  =  0,  etc., 

and  these  in  Maclauriu's  formula  give  the  series  above. 

Since  a  series  whose  terms  are  alternately  plus  and  minus 
converges  if  each  term  is  numerically  less  than  the  preceding, 

the  series  converges  for  x  =1,  whence  tan~'l=  45°  =  -,  and  we 
have 

whence  the  value  of  tt. 

9.  sin  (x  -\-y)  =  sin  x  cos  y  -f-  cos  x  sin  y. 

This  being  a  function  of  the  sum  of  two  variables,  we  use 
Taylor's  formula. 


APPLICATIONS   OF   SUCCESSIVE  DIFFERENTIATION.      79 

Making  y  =  0,  f(x)  =  sin  x,  whose  successive  derivatives  are 

f{x)  =  cosx,  f"(x)  =  —  sinx,  f"'{x)  =  —  cosx,  /'''(a.')=  sina;, 

and  so  on  in  sets  of  four.     Hence,  substituting  in  Taylor's 

formula, 

....  .       V^  tf       .       V*  ff 

^vi\{x-\-y)  =  sin 0;+ cos  x-y  —  sm a;-^  — cos x'^  +  sm a;"^ +cos x.-p •  •  • 

=sin^|l-|'  +  g-...|+eos.{,,y-|  +  |-...| 

=  sin  cc  cos  ?/ 4- cos  a;  sin  ?/  (Exs.  2  and  3). 

10.  cos  (x  +y)  =  cos  a;  cos  ?/  —  sin  a;  sin  y. 

11.  sin  [x  —  y)  =  sin  x  cos  y  —  cos  x  sin  y. 

12.  cos  (x  —  y)  =  cos  x  cos  y  +  sin  x  sin  y. 

13.  Deduce  the  Binomial  formula  by  Taylor's  theorem  from 
(a; +  ?/)•». 

Making  y  =  0,  fix)  =  a;"*,  whose  successive  derivatives  are 

/'  (a;)  =  mx'"""',  f"(x)  =  m  {m  —  V)  x'^~-,   etc., 
hence       (a;  +  ?/)•"  =  a-"*  +  mx'^-^y  +  m{m  —  l)x'"'^'^  +  etc. 

Main  T  ^       I  I      »^  O  3/        ,      O  t*/  O  it/ 

^[2       |4  ^  |5        |6 

1  2  ^  1 4         I G         J 

IG.  tana;  =  xH 1 . 

3       15 

17.  secaj=l +  -  +  —  •••• 

2       24 

18.  -^  =  l  +  a^  +  a;2  +  ?|!-f^.... 
cos  a;  3        2 

19.  a;V=x-2-|-ar^-|--  +  -.... 

[2      [3 


80  THE  DIFFERENTIAL   CALCULUS. 


20.  e"'"'=l  +  x2  +  -. 
3 


.v3  T  ^■t 


21.  etan-'x  =  i  +  a;  +  -- --—.... 
2       6       24 

feav    -r^;  t,a    -r      1^      2a^      Sar*      4x* 

23.  a'-^^  =  a=^|l  +  loga.2/4-log==«|'  +  log«a^...|. 


y , ^ 


24.  sin~'(a;4- ?/)  =  sin~'.r  H " f- 

[3  (1-a^)' 

25.  (a^  -  eV)  J  =  a  -  —  -    ^'^*  ^  ^"^^ 


2a      2.4a'''      2.4.6a° 

73.   Failing  cases  of  Madaurin's  and  Taylor's  formulce. 

It  has  been  seen  that  the  above  formulae  often  lead  to  diverg- 
ing series  and  therefore  fail.  The  following  exceptions  are  also 
to  be  noted. 

Since  the  proof  that  the  formulae  develop  any  function  de- 
pends upon  the  condition  that  the  derivatives  of  the  functions 
are  continuous,  no  one  of  them  becoming  infinite  for  a  finite 
value  of  the  variable,  if  log  x  be  the  function,  whose  first  deriv- 
ative f'{x)  =  -  becomes  oo,  as  do  all  the  succeeding  derivatives, 

when  x  =  0,  the  coefficients  /'(O),  /"(O),  etc.,  of  Maclaurin's 

formula  become  infinite,  the  series  has  no  determinate  value, 

and  log  X  cannot  be  developed  in  powers  of  x.     The  same  is 

^     I 
true  of  x",  a',  cosec  x,  cot  x,  etc. 

Again,  from  (x-\-y-\-ay,  we  have,  for  y  =  0,  f(x)  =  {x  +  a)-, 
whence  f'(x)  = ->  which  is  finite  for  all  values  of  x 


2{x  +  a) 


except  x  =  —  a.     For  this  value  of  x,  f'(x)  =  co,  as  are  all  tlie 


APPLICATIONS   OF    SUCCESSIVE   DIFFERENTIATION.      81 

successive  derivatives.  Hence  the  coefficients  f'{x),  f"(x), 
etc.,  of  Taylor's  formula  become  infinite  for  x  =  —  a,  and  the 
function  (x  +  y-\-ay  can  be  developed  in  powers  of  y  for  all 
values  of  x  except  x  =  —  a. 


Evaluation  of  Illusoi'y  Forms. 


74. 


The  form  ^• 


It  frequently  happens  that  for  a  particu- 


0 


Thus 


lar  value  of  the  variable  a  function  assumes  the  form  — 

Sill   3y  0  • 

=  -  when  X  =  0.     How  is  this  result  to  be  interpreted  ? 

X        0 

Let  X,  y,  be  the  coordinates  of  P,  x  and  y  being  functions  of 
z,  and  let  MN  be  the  curve  the  coordinates  of  whose  points  are 
the  simultaneous  values  of  x  and  y  as   2 

changes.  Then  -  =  ^'  Since  by  hypoth- 
esis X  and  y  become  zero  for  some  value  of 
«,  the  curve  MN  passes  through  the  origin. 
Let  a  be  the  value  of  z  which  renders'  x  and 
y  zero.     Then  as  z  approaches  a,  x  and  y 

approach  zero,  and  P  approaches  0,  so  that 

y 
the  value  of  -  when  z  =  a  is  the  limit  of 

X 

tan  <f>,  (f>  being  the  angle  which  the  secant  makes  with  X.  But 
the  limit  of  tan  </>  as  P  approaches  0  is  tau^,  OT  being  the 
tangent  at  0 ;  hence 

tan^  =  ^^" 


Fig.  II. 


~1     = 

Xjs=a 


Therefore,  to  find  the  value  of  ■}:     ,  we  find  that  of  '  ,^     , 
since  these  are  equal. 


/'(«) 


0 


If  TTT"^  is  also  j^j  then  since /'(z)  and  <f>'{z)  may  be  regarded 

as  new  lunctions  of  z  whose  ratio  is  -  when  z  =  a, 

0  ' 


82  THE   DIFFERENTIAL   CALCULUS. 

•      <^'(a)-</."(a)' 
and  so  on  indefinitely.    Hence 

To  evaluate  a  function  which  assumes  the  form  -  for  a  par- 
ticular value  of  the  variable^  form  the  successive  derivatives  of  its 
numerator  and  denominator  and  substitute  in  them  the  particular 
value  of  the  variable,  continuing  the  process  till  a  jiair  is  found 

whose  ratio  does  not  become  -• 

0 

Examples.     Find  the  value  of : 

-,    sin  a;      ,  ,^ 

1.  when  cc  =  0. 


X 

1. 


f{x)  _cosa; 


1  —  cos  X      ■,  rt 
when  x  —  Q. 


f'(x)  _  sin  x' 
<l>'{x)  ~  2x 


_  ^  .   /"(*)  _  cos  x' 
o~0'   <i>"{x)  ~     2 


6*  —^  6~* 

3.  when  a;  =  0.  Ans.  2. 

log(H-x) 

/7*  ■        fi"^  ft 

4. when  a;  =  0.  Ans.  log-- 

X  b 

5.  when  x  =  l.  Ans.  -• 

X"  —  1  ri 

6.  -^ —         — ^^^^  when  x  =  3.  Ans.  ^. 
x^  —  X-  —  ox  —  3 

The  successive  differentiations  will  be  facilitated  by  evalu- 
ating a  factor  in  any  result  when  possible.  Thus : 


APPLICATIONS   OF   SUCCESSIVE  DIFFEKENTIATION.      83 


7. 


X  —  sill    '  X 

sin^a; 


when  cc  =  0. 


fix')  V1-X--1         _ 


Vl-x=^-l 


'^\^)      3  sin^  it- cos  a;  Vl  -  »-      cosa;Vl-a^  Ssiir'a; 

the  first  factor  of  which  becomes  1  when  cc  =  0.  Proceedmg 

fUx\                                1  X 

with   the   second   factor,   •  „ )  [  = 


-,  the 


6  cos  x*  Vl  —  V?  sin  a; 
first  factor  becoming  —  \  when  a;  =  0.     From  Ex.  1  the  value 
of  the  second  factor  when  a;  =  0  is  1.     Hence 


X  —  sin~'  X 
sin^  X 


"""^     --^  when  x=\. 


9. 


ar^-3a;  +  2 
x^-Ca^'  +  Saj-S 


when  x=-\. 


Ans.  0. 


^ns,  00. 


10.  I : —  when  x  — 


11. 


log  sin  X 

Vajtana) 
(e^-1)^ 


when  a;  =  0. 


Ans.  a  log  a. 


Ans.  1. 


Write   in   the   form  ^ ^ —  and  evaluate   the 

\e^  — 1     X     e'  —  l 

factors  separately. 


^  o    tan  X  —  sin  x      ,  a 

12. when  x  =  0. 


13. 


af* 

tan  a;  —  a; 

X  —  sin  x 


when  a;  =  0. 


Ans. 


Ans.  2. 


14. 


e'  —  e~ 


{e'-iy 


when  a;  =  0. 


Ans. 


84 


THE  DIFFERENTIAL   CALCULUS. 


75.  The  form  —    When  f{x)  and  <j){x)  both  increase  indefi- 
nitely as  X  approaches  a,  then  •■  ^^-  =  — . 

<l>{x)     x=a       CO 
1 


/(^•)J 


<i>{x) 


0 


=  ^-     Hence  the  form  —  can  be  reduced  to 
0  00 


the  form  -  and  treated  as  already  explained.     Thus 


sec  3  a; 


_  00    -r>  i.   sec  X  _  cos  X 
\     00  sec  3  a;  1 


cos  3  a; 
Hence,  by  the  process  already  established, 


cos  3  a:"! 

■J; 


cos  a; 


f{x)^  -3 sin 3a;' 
</)'(a;)         —sin  a; 


=  -3. 


This  transformation,  however,  will  not  always  be  successful 
unless  the  terms  become  infinite  because  of  a  denominator  in 
each  which  becomes  zero.     Thus,  in  the  above  example,  sec  a; 

becomes  infinity,  because  it  may  be  written whose  denom- 

cosa: 

ioator  becomes  zero. 


II.   Since  Z(4  =  ^ 

/(^)J 
process  of  Art.  74,  we  have,  when  x  =  a, 

<l>'{x) 


=  -,  if  we  treat  the  latter  by  the 
0  ^ 


or 


<t>ix) 

L<^(a;)J 

'^'{^) 
/'(^) 

f(x)  c^'(a;) 

<^(^)  /'(^) 


(1) 


(2) 


APPLICATIONS   OF    SUCCESSIVE   DIFFEKENTIATION.      85 

whence  -^^  ^  =  -^ }  (  ; 

cfy{x)        <\>\X) 

and  the  form  —  can  be  treated  directly  in  the  same  way  as 

the  form  — 
0 

Since  all  the  derivatives  of  a  function  which  becomes  oo  for 

0.  finite  value  of  the  variable  also  become  infinite  (Art.  56),  this 

process  would  appear  to  lead  to  no  result   except  when  the 

f(x)     f'(x) 
given  value  of  the  variable  is  infinite,    ,,,  .,    ,,,,  {,  etc.,  be- 
°  '  cf>  (x)    <i>  {xy 

coming  in  turn  —    This  is  true,  but     ,        may,  by  changing 

f(x) 
its  form,  be  more  easily  evaluated  than  . ,   , .     Thus 


logx 
x 


CO 

co' 


1 


^ .       =      m  which  also  becomes  —    for  ic  =  0,  but  it  may 

</)'(»)        _1'  CO  '  -^ 

readily  be  put  under  the  form =  —  x]o  =  0. 

In  any  case,  therefore,  when  a  function  assumes  the  form  — 

for  a  finite  value  of  the  variable,  it  is  necessary  to  transform 
either  the  function  (I.),  or  some  one  of  the  derived  functions 
(II.)  so  that  it  will  not  assume  this  form  for  the  given  value 
of  the  variable. 

f(x') 
III.   If  the  true  value  of     \  i  when  x  =  a  i&  zero  or  infinity,  equation 
<p{x) 

(1)  is  satisfied  independently  of  equation  (2),  and  it  would  therefore  ap- 
pear that  in  such  cases  the  latter  is  not  necessarily  true.     That  equation 

(2)  holds,  however,  when  the  true  value  of  -'S^  is  zero  or  infinity  may 
be  shown  as  follows : 


86  THE   DIFFEllENTIAL   CALCULUS. 

First.  Let  -')■'  =  0  when  x  —  a.     Then,  if  c  be  any  finite  quantity, 
<p{x) 

^-^  +  c  is  finite,  and  to  tiiis  function  tlie  process  of  II.  applies  since  it 
holds  whenever  the  function  does  not  become  zero  or  infinity.     Hence 

or  •  ,,  {  =  0  when  ^^^  =  0,  and  the  process  therefore  gives  the  true 
value. 

Second.  Let  '-—-^  =  qo  when  x  =  a.     Tlien  ~~-  =  0.     Hence,  by  the 
<t>(.^)  /(«) 

preceding,  ^^  -'  =  — ^,  or  •')  (  =;^^^,  and  the  process  holds  m  this 

/(o)     /(«)         ^(«)      <?>'(«) 
case  also. 


Examples.     Evaluate : 

^     tan  cc      1  TT 

1.  w^nena;  =  -- 

tan  5x  2 

sin  a; 


tana;        cos  a;        sin  a;  cos  5  a; 


tan  5  X      sin  5  a;      sin  5  x  cos  a; 
cos  5  a; 

When  a;  =  7TJ  the  first  factor  is  1,  and  the  second  factor  be- 

0 
comes  -  •     Evaluating  the  latter  by  Art.  74;  we  find 


tan  X  " 
tan  5  a; 

irX 


.  =  5. 

2 


sec"^ 

when  x  =  l. 


log  (1  -  x) 


TT  TTX   .  TTX  TTX 

-  sec  —  tan  —       tan  -r: 
f'{x)     2         2  2        ^^^  2 


<b'(x)  1  irX 

^  ^  ■' —  cos  — 

1-x  2 

1-X 


APPLICATIONS   OF   SUCCESSIVE  DIFFEKENTIATION.      87 


When  a;  =  1,  the  denominator  becomes  ->  and  differentiating 
once  we  find  its  value  to  be  —  1.     Hence 


sec 


~2 


log(l-a;). 


=  GO. 


loga; 


when  X  =  oc.  Ans.   0,  or  go,  as  »i  >  0  or  n  <  0. 


4.  ^Qg^^"^^^whenx  =  0. 
log  tan  X 


Ans.   1. 


76.  The  form  ox  qo.   When,  for  x=a,f(x)=0  and  (l>(x)  =  <x>, 
f{x)  c}>{x)  =Ax)^  =  I,  or  f{x)  ^(x)  =  -j-  <l>{x)  =  ^. 

Hence,  by  introducing  the  reciprocal  of  one  of  the  fac- 
tors, the  function  may  be  reduced  to  one  of  the  two  forms 
-,  ~,  as  is  most  convenient,  and  treated  as  before. 

0      GO 


Examples.     Evaluate : 

1.   (1  —  x)  tan^  when  x  =  l. 

z 

TtX       \—x 
(1  —  ic)  tan  -^  = 

cot  — 

2. 

Hence,  by  Art.  74, 
\-x- 

.irX 

cot  — 

2_ 


TT  o  TTX 

■  -cosec-  — 
2  2. 


2.  e  "(1  — logic)  when  a;  =  0. 


-Li  1  X         1— logic"!  GO 

e  '(1  -  logx)  = j-^-    =  -. 

-       Jo     °^ 


88 


THE   DIFFERENTIAL   CALCULUS. 


Hence,  by  Art.  75, 


l-loga;1      Q^ 


3.  e* sin  —  when  x  =  cc. 
e' 


.     « 

_  .     a  e* 

e''  sm  —  = 

e'       e' 


Alls.  ft. 


4.  (a''— l)a;  when  a;  =  oo. 


^>i,s.  log  a. 


77.   The  form  <x>  —  cio.    When,  for  x  =  a, 
/(x)  =  oo  and  <}}(x)  =  cc, 


f(x)-^(x)  =  -^ ^ 


^{x)      f(x)      0 
1  ~0' 

and  may  be  treated  when  thus  transformed  by  Art.  74. 
Examples.     Evaluate : 


log  X     log  x 


when  a;  =  1. 


Here      /(a^)  =  rrrr.'    '^(^)  = 


log  a;  log  X 

1  1        log  a;     1 

—^ log  X 

Hence      </'(^)     /(^)^^ ^1^^ 

1  1  loga;_ 

f{x)c{y(x)  X 


_0 


los^a; 


and,  by  Art.  74, 
1-x 


logaj 


-1 


=  -1. 


APPLICATIONS   OF   SUCCESSIVE  DIFFERENTIATION.      89 

2.  — ^  when  x  =  \. 

x  —  \      log  X 

Transforming,  as  above,  we  obtain 

0 


Hence 


a;  log  eg  —  x-\-\ 
X  log  X  —  log  X 

X  log  x  —  x-\-\ 


0 


loero; 


X  log  X  —  log  a; 
3.  sec  X  —  tan  x  when  x  = 


loga;  +  1 

a;ji 


x  +  1 


This  may  be  transformed  as  above ;  or,  more  directly, 


sec  X  —  tan  x  = 


sin  X      1  —  sin  x 


cos  X     cos  a; 


cos  a; 


Ans.  0. 


when  a;  =  1. 


a^-1      x-\ 

This  may  be  transformed  as  above ;  or,  reducing  to  a  common 
denominator, 

2  1  2a;-a^-l 


or^-l 


1      a:;^  —  a.-^  —  a;  +  l 


^       2-2a;      1 
1     3ar-2a;-lJ, 


-2 


6a; -2 


5.  a;  tan  a;  —  -  sec  a;  when  a;  =  -■ 
2  2 


^7lS.    —  1. 


78.  The  forms  qc°,  l",  0°.  The  logarithm  of  a  power  may 
assume  an  illusory  form  under  the  following  circumstances. 
Let  2*  be  the  function.     Passing  to  logarithms, 

log  2^=2/ log  2, 

which  becomes  0  x  oo  when  i^""^'^.,     ~'     **     ~     „ 

(y  =  0and2  =  co,    .-.  2"=  co  , 

and  which  becomes  oo  x  0  when  y  =  oo  and  z  =  \,  .".  z"  =  1°°. 
The  logarithms  of  such  functions  may  therefore  be  evaluated 


90 


THE  DIFFERENTIAL  CALCULUS. 


as  in  Art.  76,  and  thus  the  values  of  the  functions  themselves 
are  readily  obtained. 

The  functions  0*  and  oo°°  do  not  give  rise  to  illusory  forms, 
as  may  be  seen  by  passing  to  logarithms,  the  logarithms  in 
both  cases  being  infinity. 

Examples.    Evaluate : 
1.  (-4-11  when  x=cc  . 


M 


Putting  v=(--\-l],   logv  =  a;logf-+l 


a      X 


ccra-\-x        ax 
~  a  +  x 


1 


1' 


•.  v  =  e" 


log^l 

1 

x 


2.  M  -f  -2  j  =  V,  when  x=yo  .      Ans.  log i;  =  0,  .'.  v  =  e^=  1. 


3.  (sin  a;)**"' when  a;  =  - ■ 

4.  af  when  a;  =  0. 


-u  =  af  •  log  v=-x  log  X 


Hence  v  =  1. 


loga^ 

X 


Ans.  1. 


— 1  =  -a^]o=0. 


5.  x^'"  when  x=i 

,  log  a;" 

log  v  = 


1-x 


=  -1, 


G.   (sin x)"^''  when  x  =  0. 

7.   (cot  a:)*''"'  when  x  =  0. 

(cotx)«'"^=-(-^5^-^ 
^         ^  (sina;)'""'=_ 


=  -^„,or(Ex.6),l. 


A71S.  1. 


APPLICATIONS   OF   SUCCESSIVE  DIFFERENTIATION.      91 

8.  (sin a;)'""*  when  a;  =  0.  Ans.   1. 

1 

9.  (i  +  ax)'  when  a;  =  0.  Ans.    e". 

10.  a.'^-i  when  a;  =  0.  Ans. 


ins.   1.  I 


Maxima  and  Minima  Values  of  a  Function  of  a  Single  Variable. 

79.  The  value  of  a  function  is  said  to  be  a  maximum  when  it 
is  greater  than  its  immediately  preceding  and  succeeding  values, 
and  a  minimum  tvhen  it  is  less  than  its  immediately  preceding 
and  succeeding  values. 

By  greater  and  less  values  are  meant 
algebraic  values.  Thus,  if  MN  be  the 
locus  of  y=f(x),  and  mn  is  greater 
than  the  immediately  preceding  and 
succeeding  ordinates,  mn  is  a  maxi- 
mum value  of  y.  Similarly  pq  is  a 
minimum  value  of  y.      It  is  evident 

that  for  increasing  values  of  x,  y  diminishes  after  passing 
through  a  maximum  value,  and  cannot  therefore  have  a  second 
maximum  value  without  first  passing  through  a  minimum ; 
or  maxima  and  minima  values  occur  alternately.  From  the 
definition  it  is  also  evident  that  a  maximum  value  is  not 
the  greatest  possible  value,  nor  a  minimum  the  least  possible 
value,  of  a  function. 

80.  Condition  of  a  maximum  or  minimum  value. 

For  increasing  values  of  x,f(x)  is  increasing  before,  and  de- 
creasing after,  a  maximum  value.  Hence  (Art.  22),  f'(x)  is 
positive  before,  and  negative  after,  a  maximum  value  of  f{x) ; 
or  the  first  derivative  of  the  function  changes  sign  from  plus 
to  minus  as  the  function  passes  through  a  maximum  value. 

Similarly,  a  function  decreases  as  it  approaches  a  minimum 
value  and  increases  after  such  value ;  or  the  first  derivative 
changes  sign  from  minus  to  plus  as  the  function  passes  through 
a  minimum  value. 


92 


THE  DIFFERENTIAL   CALCULUS. 


Since  in  either  case  the  first  derivative  changes  sign,  it  must 
pass  through  zero  or  infinity.  Hence,  every  value  of  x  ivhich 
renders  f{x)  a  maximum  or  a  minimum  is  a  root  of  f'{x)  =0, 
or  off  (x)  =Go. 

It  is  to  be  observed  that  the  essential  characteristic  of  a 
maximum  or  minimum  value  of  the  function  is  c.  change  of  sign 
of  its  first  denvative.  Now  Ci  quantity  may  become  zero  or 
infinity  without  changinrj  sign  5  hence  the  roots  of /'(a;)=0 
and  f'{x)  =  00  are  called  critical  values,  and  must  be  separately 
examined ;  only  those  for  which  f'{x)  changes  sign  can  corre- 
spond to  maxima  or  minima  values  of  the  function. 


Fig.  13, 


81.  Geometric  illustrations.  Since  y=f{x)  is  the  equation 
of  some  locus,  and  f'{x)  is  the  slope  of  the  locus  at  any  point, 
the  foregoing  remarks  admit  of  the  following  illustration : 

In  Fig.  13,  Pm  being  a  maximum 
value  of  y,  for  increasing  values  of  x 
the  angle  made  by  the  tangent  with 
X  is  acute  before,  and  becomes  obtuse 
after,  the  maximum  value  ;  hence  the 
tangent  of  this  angle,  which  is  /'  (x) , 
is  positive  before  and  negative  after 
this  value.  At  P  the  tangent  is  par- 
allel to  X,  and  its  slope  is  therefore 
zero. 

In  Fig.  14,  Pm  being  a  minimum 
value  of  y,  the  angle  made  by  the  tan- 
gent with  X  is  obtuse  before  and  acute 
after  the  minimum  value  of  y,  the  slope 
changing  from  minus  to  plus,  and  pass- 
ing through  zero  as  before. 

In  Fig.  15,  although  the  tangent  at 
P  is  parallel  to  X  and  therefore  /'  (x) 
is  then  zero,  the  angle  is  obtuse  both 
before  and  after  the  value  x  =  Om  and 
does  not  change  sign ;  hence  Pm  is 


Fig.  14. 


r 

<^       Fig.  15. 

X.P\ 

\ 

V-s      >^<-^ 

t 

>               n 

\       - 

APPLICATIONS   OF   SUCCESSIVE   DIFFERENTIATION.      93 


neither  a  maximum  nor  a  minimum  value 
of  y. 

The  change  of  sign  of  f'{x)  from  +  to 
— ,  and  from  —  to  +,  in  passing  through 
infinity  is  shown  in  Fig.  16,  the  tangent  at 
P  being  perpendicular  to  X  and  its  slope 
infinity. 


82.   Examination  of  the  critical  values  when  /'(a?)  =  0. 

Since /'(x)  changes  sign  from  +  to  —  as /(a;)  passes  through 
a  maximum  value,  it  is  a  decreasing  function,  and  its  first  deriv- 
ative/"(a;)  must  be  negative  (Art.  22). 

Also,  since  f'{x)  changes  sign  from  —  to  +  as  f(x)  passes 
through  a  minimum  value,  it  is  an  increasing  function,  and  its 
first  derivative /"(a;)  must  be  positive. 

Hence,  to  examine  f(x)  for  maxima  or  minima  values,  observe 
whether  f" {x)  is  negative  or  positive  for  critical  values  of  x,  that 
is,  for  values  derived  from  the  equation /'(a;)  =  0. 

As  the  second  derivative  may  become  zero  for  a  critical 
value  of  X,  the  above  test  may  fail.  To  provide  for  such  case 
we  have  the  following  more  general  rule. 


83.  Let  y  =f(x),  and  y^  =f(xi)  in  which  x^  is  the  value  of 
X  which  renders  y  =  y^=.a.  maximum  or  a  minimum. 

Let  y'=f(Xi  —  h)  and  y"  =f(xi  +  h)  be  the  values  immedi- 
ately preceding  and  succeeding  the  maximum  or  minimum 
value  2/1,  Xi  —  h  and  x^  +  h  being  the  corresponding  values  of  x. 
Developing  y'  and  y"  by  Taylor's  formula,  we  have 

y'  =f{x,  -  h)=f{x,)  -  f\x,)h  +  fix,)  'I 

If 


y"  =f{x^  +  h)  =f{x,)  +  f\x,)h  +  f"(x,) 


+f"'ix^)^+f"(^i)~ 


94  THE   DIFFERENTIAL   CALCULUS. 

But  f(x)  I  =  2/1,  and  since  Xi  corresponds  to  a  maximum  or  a 
minimum, /'(a;i)  =0.     Hence,  transposing, 

2/'  -2/i=/"(^i)|-/"'(^i)|'+/^(a:i)^'-,  (1) 

y  -  y,  =/"(a^i)|  +r{x,)  1 4-/n^i)  I  -.  (2) 

Now  the  signs  of  the  second  members  of  (1)  and  (2)  will  be 
those  of  their  first  terms,  that  is  of  /"(a^i),  if  h  be  taken  suffi- 
ciently small ;  and  since  h  approaches  zero  as  the  function 
approaches  its  maximum  or  minimum,  we  are  at  liberty  to 
make  h  as  small  as  we  please.  Hence  if /"(cKj)  is  positive,  the 
first  members  are  positive,  and  both  ?/'  and  y"  greater  than  y-^, 
which  is  therefore  a  minimum;  while  if/"(.Ti)  is  negative, the 
first  members  are  negative,  both  y'  and  ?/"  are  less  than  y^,  and 
2/1  is  a  maximum.  This  accords  with  what  has  already  been 
said. 

If /"(a^i)  is  zero,  then 

y"-y,=    /'"(a;0^+/^(a;Og-, 

in  which,  whatever  the  sign  of  /'"(Xj),  the  first  members  will 
have  opposite  signs,  and  y'  and  y"  cannot  both  be  greater  than 
2/1,  nor  both  less.  Hence  neither  a  maximum  nor  a  minimum 
can  exist  unless  f"'{Xi)  =  0,  If  this  condition  be  fulfilled,  there 
will  be  a  maximum  or  a  minimum  according  as  /'"'(it'i)  is  nega- 
tive or  positive.     We  have  therefore  the  following  rule : 

To  determine  whether  a  function  has  maxima  or  minima  val- 
iies,  form  its  first  derivative  and  place  it  eqiial  to  zero.  Tlie  roots 
of  this  equation  contain  the  values  of  the  variable  which  correspond 
to  either  maocima  or  minima  values  of  the  function.  Find  the  first 
derivative  which  does  not  become  zero  for  one  of  these  critical  val- 
ues of  the  variable.     If  this  derivative  is  of  an  odd  order,  there  is 


APPLICATIONS   OF   SUCCESSIVE   DIFFERENTIATION.      95 

neither  a  maximum  nor  a  minimum;  if  of  an  even  order,  the 
function  is  a  maximum  or  a  minimum  according  as  the  derivative 
is  negative  or  positive. 

Each  critical  value  must  of  course  be  examined  in  turn. 

Illustration.  Examine  o(^  —  5x*  +  5x'  +  l  for  maxima  and 
minima  values. 

f\x)  =5x*-20a^  +  15x'  =  5a^(a^-4:X  +  3)  =  0. 

The  roots  of  this  equation  are  a;  =  0,  a^  =  1,  x  =  3. 

f"{x)  =  20  ar^  -  GO  a^  +  30  a;  =  10  a;  (2  .r-'  -  G  a;  +  3). 

Substituting  x  =3,  f" (x)  = -\- 90  ;  hence  x  —  3  renders  the 
function  a  minimum,  and  substituting  this  value  of  x  in  the 
function  we  find  f{x)  =  —  2G,  the  minimum. 

Substituting  a;  =  1,  f"{x)  =  —  10 ;  hence  a;  =  1  renders  the 
function  a  maximum,  which  we  find  to  be  2. 

As  f"(x)  =  0  for  a;  =  0,  we  form  f"'(x)  =GOx^-  120  a.'  +  30, 
which  does  not  vanish  for  a;  =  0  and  is  of  an  odd  order.  Hence 
a;  =  0  corresj)onds  to  neither  a  maximimi  nor  a  minimum. 

84.  Abbreviated  processes. 

I.  Since  the  essential  characteristic  of  a  maximum  or  mini- 
mum value  of  a  function  is  a  change  in  the  sign  of  its  first  de- 
rivative, it  will  be  sufficient,  when  possible,  to  observe  whether 
for  a  critical  value  of  the  variable  such  change  actually  takes 
place.  Thus,  from  {x  —  a)*  +  b,  f'{x)  =  4  (a;  —  a)''  =  0,  the 
critical  value  being  x  =  a.  Now  in  passing  through  x=  a, 
f'{x)  changes  sign  from  —  to  +  ;  hence  x=a  renders  the 
function  a  minimum,  namely  b.  Again,  from  (x  —  ay  -\-  b, 
f(x)  =  3  (x  —  a)^  =  0,  which  cannot  change  sign  for  any  value 
of  x;  hence  the  function  has  no  maxima  nor  minima  values. 

II.  Since  if  yl  is  a  constant  factor,  Af{x)  increases  and 
decreases  with  /(a;),  a  constant  factor  may  be  omitted  in  the 
search  for  maxima  or  minima  values. 


96  THE   DIFFERENTIAL   CALCULUS. 

III.  Since  ±A  -\-f{x)  increases  and  decreases  with/(a;),  we 
may  substitute /(x)  for  ±A-\-f(x)  in  searching  for  maxima  or 
minima  values.  If  A—f{x)  is  the  given  function,  we  may 
substitute  f{x),  provided  we  reverse  the  conclusions,  as 
A  —f(x)  increased  when  f(x)  decreases,  and  decreases  when 
f(x)  increases. 

IV.  Since  — -  decreases  as  f(x)  increases,  and  conversely, 

the  reciprocal  of  the  function  may  be  substituted  for  the  func- 
tion, provided  the  conclusions  are  reversed. 

V.  Since  log  [/(a^')]  increases  and  decreases  with  f{x),  the 
number  may  be  substituted  for  the  logarithm  of  the  number 
in  the  search  for  maxima  and  minima  values. 

VI.  If  f{x)  is  positive,  [/(a;)]"  is  also  positive,  and  there- 
fore increases  and  decreases  with  f(x)  ;  or  any  power  of  a  posi- 
tive function  may  be  substituted  for  the  function.  If  f(x)  is 
negative,  [/(>>/•)]"  will  have  the  same  sign  as  f{x)  if  n  is  odd, 
but  the  opposite  sign  if  n  is  even ;  or  any  power  of  a  negative 
function  may  be  substituted  for  the  function,  provided  the 
conclusions  are  reversed  if  n  is  even. 

We  are  thus  enabled  to  omit  the  radical  sign  in  the  search 
for  maxima  and  minima  values  of  any  positive  radical ;  also 
when  the  radical  is  negative,  if  we  reverse  the  conclusions. 

Examples.  Examine  the  following  functions  for  maxima 
and  minima  values. 

1.  ^-3-90^  + 15a' -3. 

Omitting  the  constant  term  (III.,  Art.  84), 

f{x)=x^-9x^  +  15x. 

f'{x)  =  Sx^  —  18  a;  -j- 15  =  0,  whence  the  critical  values 

x=  5,  X  =  1. 

f'{x)  =  6a;  -  18,  which  is  12  for  a;  =  5  and  - 12  for  x  =  l. 
Hence  x  =  5  renders  the  function  a  minimum,  and  x  =  l  ren- 


APPLICATIONS   OF   SUCCESSIVE   DIFFERENTIATION.      97 

ders  it  a  maximum.  Substituting  x  =  5  and  x  =  1  in  the  func- 
tion, its  minimum  and  maximum  are  found  to  be  —  28  and  4, 
respectively. 

2.  b  +  c(x  —  d)\ 

f(x)  =  (x-a)  3  (Art.  84,  II.  and  III.). 

f'{x)  =  ^(x  —  ay  =  0,  whence  x  =  a;  and  as  f'(x)  changes 
sign  from  —  to  +  for  increasing  values  of  x  as  x  passes 
through  a,  &  is  a  minimum  value  of  the  function. 

3.  x^-5x*-i- 5x^-6. 

4.  Examine  the  circle  y^-\-x^=R^  for  maxima  and  minima 
ordinates. 

The  function  to  be  examined  is  y  =  ±  Vii'  —  o?.  Omitting 
the  radical  (Art.  84,VI.),/(x)  =R--  x',  whence /'(a;)=  -2x=0, 
OT  x  =  0',  and  as /'(a*)  changes  sign  from  -f  to  —  as  a;  passes 
through  0,  a;  =  0  corresponds  to  a  maximum.  If  we  take  the 
negative  value  of  the  function,  then,  in  omitting  the  radical,  we 
raise  the  function  to  an  even  power  and  must  reverse  the  con- 
clusion ;  hence  when  y  is  negative,  x  =  0  corresponds  to  a  mini- 
mum. 

5.  (x-iy{x-\-2y. 

f\x)=A{x-iy(x-h2y  +  s(x-iy(x  +  2y 

=  (x  —  ly  {x -\- 2y  (7  X -{- 5) ;  whence  the  critical  val- 
ues X  =  1,  X  =  —  2,  X  =  —  ^. 

Since /'(a;)  is  —  if  x  is  a  little  less  than  1,  and  -f-  if  a;  is  a 
little  greater  than  1,  it  changes  sign  from  —  to  -}-  as  x  passes 
through  1 ;  hence  a;  =  1  corresponds  to  a  minimum. 

f"(x)=3ix-iy(x-{-2y{7x-{-5)+2{x-iy{x-\-2){7x  +  5) 

-^7(x-iy(x  +  2y 

=  (x-iy(x  +  2)\3(x  +  2){7x+5)+2(x-l)(7x+5) 

-^7(x-l)(x+2)\ 
=  6{x  -  ly  (x  -f  2)  (7x'-\-  10a;  +  1). 


98  THE   DIFFERENTIAL   CALCULUS. 

When  ic  =  —  f ,  the  first  two  factors  are  positive,  and  the  sign 
will  depend  upon  that  of  the  third  factor,  which  is  —  for  a;=  —  |; 
hence  x  =  —  ^  corresponds  to  a  maximum. 

Since/"(x)  =Oforx=  -2,f"'{x)  =6{x-iy  (7 ar'+lOrc+l)  + 
other  terms  which  contain  (a;  +  2),  and  which  therefore  vanish 
when  cc  =  —  2,  while  the  term  6(a;  — 1)^(7 a^  +  10a; +  1)  does 
not.  Hence  f"'(x)  does  not  become  zero  for  a;  =  —  2,  and  this 
value  of  X  corresponds  to  neither  a  minimum  nor  a  maximum. 

6.  xi^  —  3x^  +  6x  +  7.     The  critical  values  are  imaginary. 

7.  Sin^  X  cos  x. 

f'(x)  =3  sin^  X  cos^ x  —  sin* x  =  3  sin^  x(l  —  sin^  a;)  —  sin* x 

=  3  sin^ X  —  4:  sin* x  =  0;  whence 
sin^a;(3  — 4sin^.^•)=0,  and  the  critical  values    are   sina;  =  0, 

sina;=-— ,  or  a;  =  0°,  a;  =  60°.     Since  f'{x)   evidently  changes 

sign  from  +  to  —  as  sin  x  passes  through  the  value  — ,  x  =  60° 

corresponds  to  a  maximum.  If  x  is  a  little  greater  or  less  than 
0°,  4sin^a/'<3  and /'(x)  is  +;  hence  a;  =  0°  corresponds  to 
neither  a  maximum  nor  a  minimum. 


8.  a+ V4ar'-2a.-3. 

Omitting  the  constant  term,  radical  sign,  and  factor  2  (Art. 
84,  III.,VI.,  II.),  we  have  2a;'-  x^;  whence /'(a;)  =  4a;  -  Sa;^  =  0, 


or  a;  =  0,  a;  =  |. 

f"(x)  =4  —  6a;,  which  is  -f  for  a;  =  0  and  —  for  a;  =  |.  Hence 
the  function  is  a  minimum  when  a;  =  0  and  a  maximum  when 
X  =  ^. 

9.  Divide  a  into  two  factors,  the  sum  of  which  shall  be  a 
minimum. 

Let  X  =  one  factor  ;  then  -  =  the  other,  and  the  function  is 

X  ' 

a;  +  -  ;  or  f'{x)  =  1  —  -^  =  0,  whence  a;=  Va,  and  the    factors 


a; '       -^  ^^  ~         x" 
are  equal. 


APPLICATIONS   OF   SUCCESSIVE   DIFFERENTIATION.      99 

10.  The  difference  between  two  members  is  a.  Prove  that 
the  greater  =  twice  the  less  when  the  square  of  the  greater 
divided  by  the  less  is  a  minimum. 

11.  Find  a  number  a;  such  that  its  x'th  root  shall  be  a  maxi- 
mum. Ans.   X  =  e. 

12.  To  determine  the  number  of  equal  parts  into  which  a 
must  be  divided  in  order  that  their  continued  product  may  be 
a  maximum. 

T  in  1       «  .  a   a    a 

Let  X  =  number  oi  parts  :  then  -  =  one  part,  and  -  •  -  •  — • 

/     \:r.  X  ^  XXX 

/ct\ 
to  X  factors  =  (  -  )  is  to  be  a  maximum. 
\xj 

log  I    ]=  x([og  a  -  log  x)  =f(x). 

/'(a;)  =  logct  — logx  — 1  =  0,  or  log  -  =  1,  whence  -  =  e,  or 
a 

X  =  -' 

e 
Arithmetically  the  problem  would  not  be  possible  unless  a 
was  a  multiple  of  e,  otherwise  x  would  not  be  an  integer.    The 
general  solution  belongs  to  the  statement :  to  find  a  number  x 

such  that  the  ccth  power  of  -  shall  be  a  maximum. 

1--^ 


13.  — ^ — r(x)= — ^^i^ 

1  + a;  tan  a;  (1+cctana;)^ 

A  maximum  when  x  =  cos  x. 

.  ,        sin  X         ,,,   ^        1  —  tan^  x 
14.  zr—: /'(^•)  = 


1  +  tan  a;  (1  +  tan  x)'' 

cos  X 
A  maximum  when  x  —  45°. 


85.   Examination  of  the  critical  values  when  /'  {x)  =  oc . 

Since,  when  f'{x)  =go  for  a  particular  value  of  x,  f"(x), 
f"'{x),  etc.,  also  become  infinity  (Art.  56),  the  function  cannot 


100  THE   DlFFEllENTIAL   CALCULUS. 

be  developed  by  Taylor's  formula,  and  the  results  of  Art.  83  are 
inapplicable.  In  such  cases  we  may  examine  the  first  deriva- 
tive directly  to  see  if  it  changes  sign  as  the  variable  passes 
through  its  critical  value. 

Examples.     1.  b+{x  —  ay. 

f'(^x)  =  ^(x  —  a)~^  = =00,  whence   x  —  a  =  0,   or 

3(x-a)^ 
x  =  a.     It  is  readily  seen  that  f'{x)  changes  sign  from  —  to 
+,  and  that  x  =  a  therefore  corresponds  to  a  minimum. 

2,    (^  +  2)« 


{x-sy 

(0^  +  2)^(0.-13) 

/w-      (^x-'sy 

y (a;)  =  Ogives  x  =  —  2  and  o;  =  13.    f'(x)=oo  gives  a;  =  3. 

o;  — 13  is  negative  if  o;  is  a  little  less  or  greater  than  3, 
while  (o;  —  3)^  is  negative  if  o;  <  3  and  positive  if  o;  >  3. 
Hence /'(a;)  changes  sign  from  +  to  —  at  o;  =  3,  which  gives 
a  maximum. 

a;  =  —  2  and  a;  =  13  may  be  examined  in  like  manner  ;  the 
latter  gives  a  minimum,  and  the  former  neither  a  maximum 
nor  a  minimum. 

3.   (^-^>^ 


(a; +  1)3 

f'(x)  =  0  gives  a;  =  1  and  x  =  5,  the  former  corresponding  to 
a  minimum,  and  the  latter  to  a  maximum.  /'  (a;)  =  oo  gives 
a;  =  —  1,  which  corresponds  to  neither. 

86.  Geometrical  Problems. 

In  the  following  problems  F=  volume,  A  =  area,  S  =  sur- 
face, and  the  substituted  function  obtained  after  omitting 
constant  factors,  radical  sign,  etc.  (Art.  84),  is  designated  by 
an  accent. 
\  1.  Determine  the  rectangle  of  greatest  area  which  can  be 
inscribed  in  a  given  circle. 


APPLICATIONS   OF    SUCCESSIVE   DIFFERENTIATION.       101 

Let  R  =  radius.     If  we  take  x,  y,  to  represent  the  half -sides 
of  the  rectangle,  then  the  equation  of  the  cir- 
cle gives  the  relation  af  -\-y^  =  It%  by  means 
of  which  we  can  eliminate  y  from  the  expres- 
sion for  the  area  A  =  4:xy,  obtaining  _         ^  _ 


4:X^B^  —  x^,  or  ^^m^af  —  x*, 

thus  reducing  the  function  to  be  examined  to  one  of  a  single 
variable.  Omitting  the  factor  4  and  the  radical,  we  have  the 
substituted  function  A'  =  i2V  —  x*,  whence 

f'{x)  =  2E^x-4:x'  =  0,  or  ic=0  and  x  =  ^- 

V2 

f"{x)  =  2R'-12x%    which   becomes    -AR'    for    x  =  —  ; 

V2 

hence corresponds  to  a  maximum.     Substituting  x  =  — ::- 

V2  V2 

in  w-  -(-  ar^  =  Rr,  we  find  y  =  — -  ;  hence  x  =  y,  the  rectangle  is 

V2 
a  square,  and  its  area  A  =  4:xy  =  2  R-. 

Before  proceeding  to  the  remaining  examples  the  student 
will  observe :  1°.  As  the  point  P  moves  from  A  to  B,  the  area 
of  the  rectangle  increases  from  0,  passes  through  its  maximum, 
and  decreases  again  to  0.  Whenever,  then,  the  conditions  of 
the  problem  are  such  that  the  existence  of  a  maximum  value 
is  clearly  seen,  it  will  be  unnecessary  to  test  the  critical  value. 
2°.  The  solution  consists  in  first  finding  an  expression  for  the 
quantity  to  be  examined,  as  4a;y  in  the  above  case.  If  this  is 
a  function  of  two  variables,  the  next  step  is  to  eliminate  one 
by  means  of  some  relation  between  them  furnished  by  the 
conditions  of  the  problem,  as  in  the  above  case  y^  -\-a?  =  R^. 
3°.  This  elimination  may  be  effected  before  or  after  differen- 
tiation. In  the  above  case  y  was  eliminated  before  finding  the 
derivative ;  but  we  might  have  proceeded  as  follows  : 

A  =  4:xy;   A'  =  xy;  f\x)  =  x^  +  y  =  0. 

ax 


102  THE   DIFFERENTIAL   CALCULUS. 

From  X-  +  y^  =  R^,  -^  = ,  hence  f'{x)  =.  —  x'- -\- y  =  (),  or 

dx         y  y 

3i?  =  y^,  as  before.     Eliminating  now  y  by  substituting  y^  =  a^ 

in  x^  +  2/"  =  R')  we  have  x  =  — -.     It  is  frequently  preferable 

thus  to  eliminate  after  differentiating. 

2.  Determine  the  rectangle  of  greatest  area  which  can  be 
inscribed  in  a  given  ellipse. 

With  the  notation  of  Ex.  1,  A  =  'ixy,  the  auxiliary  relation 
being  a^  +  ^'•'^  —  ^'^'}  ^^^  equation  of  the   ellipse.      Hence 

A  =  4:-x^a^  —  af,  A'  =  a-x^  —  x*,  f'{x)  =  2a'X  — 4:x'^  =  0,  and 

x  =  — -,  or  2x  =  a-\/2,  which,  substituted  in  the  equation  of 

V2 
the  ellipse,  gives  2y  =  by/2,  the  sides  of  the  rectangle. 

3.  Determine  the  rectangle  of  greatest  area  which  can  be 
inscribed  in  a  given  segment  of  a  parabola. 

Let  OA  =  a,  and  y  =  the  half-side  AB.    Then 
A  =  2y{a  —  x),  or,  since  y-  =  2px, 


A  =  2^2px{a  —  x)  =  2^2py/x{a  —  xy,        o 
whence 

A'  =  a^x -2ax^  +  x\   f'{x)  =  a? -4tax  +  ^x-  =  0, 

from  which  we  find  a;  =  -•      Therefore  a  —  a;  =  f  a  =  oue  side, 

and  2y  =  2^2px  =  2-yP^=  the  other  side,  and 

Y  /       4.  Find  the  cylinder  of  greatest  volume  which  can  be  in- 
scribed in  a  given  sphere. 
With  the  notation  of  Ex.  1, 

V  =  2iry-x  =  2ivx{R''-;i?), 


APPLICATIONS   OF    SUCCESSIVE    DIFFERENTIATION.      103 


whence 


V = R'x  -  x\   f\x)=  R-  -  3  ar  =  0 , 

R  2  R 

X  =  — -,  or  2  ic  =  — :::  =  altitude. 

V3  V'S 


5.  Find  the  cylinder  of  greatest  convex  surface  which  can 
be  inscribed  in  a  given  sphere. 
With  the  notation  of  Ex.  1, 


whence 


/S  =  4  -n-yx  =  4  ttx^R-  —  or, 

S'  =  R-x-  -  x\  f  {x)  =  2  R^x  -  4  x"  =  0, 

7?  - 

X  = ,  and  2  a;  =  i2V2  =  altitude. 

V2 


6.  Find  the  cylinder  of  greatest  volume  which  can  be  in- 
scribed in  a  given  ellipsoid. 

With  the  notation  of  Ex.  1,  using  the  equation  of  the  ellipse, 

F  =  2  irf-x  =  2 TT-'  x(a'  -  x^) , 
a- 

V  =  a-x  -  x\   f  {x)  =  a-  -  3  x-2  =  0, 
whence  x  =  — - ,  and  2  a;  =  ^^^  =  altitude. 


^J' 


V3 


V3 


7.  Find  the  cone  of  greatest  volume  which  can  be  inscribe 
in  a  given  sphere. 

With  the  notation  of  the  figure,  V=  y?/'^'i   ^ 

o 

but  f  =2  Rx  -  x",  hence  V=  ^  (2  Rx"  -  x"), 

V'  =  2Rx^-  x\  /'  (x)  =  4  Rx  -  3  .'B-  =  0,  or 
x  =  ^R  =  altitude. 

\      8.  Find  the  cone  of  maximum  convex  surface  which  can  be 
inscribed  in  a  given  sphere. 


Fig.  19. 


>S'  =  7r7/V;c-+y-=7rV2  /2a;-.'C-V27e^=7rV4i2V-2i2a;«, 
S'=2Rx'-  x\  f{x)  =  4Rx-3x^  =  0, 


and  x  =  ^R=  altitude. 


104  THE   DIFFEUP^NTIAL   CALCULUS. 

9.  Find  the  cone  of  greatest  volume  which  can  be  inscribed 
in  a  given  paraboloid,  the  vertex  of  the  cone  being  on  the  axis 
in  the  base  of  the  paraboloid. 

V=-y-{a-x).     Altitude  ="      (Fig.  18.) 
o  ^ 

10.  Find  the  cylinder  of  greatest  volume  which  can  be  in- 
scribed in  a  given  right  cone. 

Let  b  =  radius  of  base,  and  a  =  altitude  of  the 
cone,  and  x,  y,  those  of  the  cylinder.  Then 
V=TTX^y.    From  similar  triangles,  h:  a::x:  (a—y), 

or  y  =  -{b-x). 
b 


h 

Hence  V=~x-{h- .x) ,   V  =  bx"  -  x\  F'g-  20. 

b 

f'{x)  =  2  bx  —  o  X-  —  0,    whence  .x  =  1 6, 
and  y  z=  -{b  —  x\  =.    =z  altitude. 

11.  Find  the  cylinder  of  greatest  convex  surface  Avhich  can 
be  inscribed  in  a  given  right  cone, 

Ans.  Altitude  =  ^  altitude  of  cone. 

12.  Of  all  right  cones  of   given  convex  surface  determine 
that  one  whose  volume  is  greatest. 

If  a;  =  altitude,  y  =  radius   of  base,   V=^y-x.     By   condi- 

o 

tion,  TT?/ V^  +  y'  =  in,  a  constant.     Diif erentiating  first,  from 

V'=y'X  we  have  f'{x)  =  2yx^  -\-  ?/-  —  0.     From  iry^x-y^  =  m, 

dx 

^  = ^ Hence    f  (x)  = — =^ 1-  ?/-  =  0,    whence 

dx  x'  +  2y'  -^  ^  ^  x'  +  2f-      ''         ' 


X 


=  2/V2,  or  the  altitude  =  V2  x  radius  of  base. 


^ 


13.  Of  all  cones  whose  slant  heights    are    equal   find  that 
which  has  the  greatest  volume. 

A71S.  The  tangent  of  the  semi-vertical  angle  =  V2. 


API'LICATIONS    OF    SUCCESSIVE   DIFFERENTIATION.      105 

14.  From  a  given  quantity  of  material  a  cylindrical  vessel 
with  circular  base  and  open  top  is  to  be  made  so  as  to  have  a 
maximum  content.  Find  the  relation  between  the  radius  and 
altitude. 

Let    .X' =  altitude,    ?/  =  radius.      Then    V=Try-x,    V'  =  y-x, 

f'{x)  =  2yx-^  -{-y-  =  0.     By  condition  2iryx  +  -n-y-  =  m,  a  con- 
dx 

stant;  whence  -^= '- — ,  and  therefore 

'  ax  x  +  y 

f\^)^-^j  +  f=^^,ovx  =  y. 

15.  A  square  is  cut  from  each  corner  of  a  rectangular  piece 
of  pasteboard  whose  sides  are  a  and  6.  Find  the  side  of  the 
square  that  the  remainder  may  form  a  box  of  maximum  content. 

a-\-h  —  Va^  —  cib  +  b'' 


Ans.  The  side  = 


6 


16.  Prove  that  in  an  ellipse  referred  to  its  -centre  and  axes, 
the  product  of  the  co-ordinates  of  a  point  on  the  curve  is  a 
maximum  when  the  co-ordinates  are  in  the  ratio  of  the  axes. 

17.  A  vertical  flagstaff  consists  of  two  pieces,  the  upper 
being  a  and  the  lower  b  feet  long.  Find  the  distance  from  the 
foot  of  the  staff  at  which  the  visual  angle  subtended  by  the 
upper  segment  is  a  maximum. 

With  the  notation  of  the  figure, 

tan  6  =  — ' — ,  tan  a  =  -, 
X  x 

,.,./]       X       tan  0  —  tan  a 
tan  <^  =  tan  (0  —  a)  = 


1  +  tan  0  tan  a 
a+b      b 

-;    whence  x  =  ^b{a  -\-b) 


^      ab  +  Jr 


106 


THE   DIFFERENTIAL   CALCULUS. 


18.  Find  the  least  triangle  which  can  be  circumscribed 
about  a  given  ellipse  having  one  side  parallel  to  the  trans- 
verse axis.  Ans.  Altitude  =  36,  base  =  2  a  V3. 

19.  Find  the  parabola  of  maximum  area  which  can  be  cut 
from  a  given  right  cone,  knowing  the  area  of  a  parabola  to  be 
|3/Q.QP(Fig.  22). 

Let  AB  =  2b,   AC=a,    QB  =  x. 
Then  MQ  =  ^AQ  •  QB  =  ^/{2b-x)x, 

and  AB:  AC  .:QB:QP,QP=^^^^=—. 
"^      ^    "^  AB         2b 

2aa; 


Hence  A= I  MQ'QP: 


3  2b 


^{2b-x)x, 


A' =  2  bx^  -  X*,   f'{x)  =  6  bx-  -  4  x'' 
whence  x  =  QB  =  |  b 


0; 


Hence   QP  =  — 
2b 


=  fa,  or  the  area  of  the  parabola  whose  axis  is  f  the  slant 
lieight  of  the  cone  is  a  maximum. 

20.  Assuming  that  the  work  of  driving  a  steamer  through 
the  water  varies  as  the  cube  of  her  speed,  find  her  most 
economical  rate  per  hour  against  a  current  running  c  miles 
per  hour. 

V  =  speed  of  steamer  in  miles  per  hour. 
av^     =  work  per  hour,  a  being  constant, 

V  —  c  =  actual  distance  advanced  per  hour. 


Let 
Then 
and 

Hence 


v  —  c 


=  work  per  mile  of  actual  advance. 


Ans.  v  =  ^c. 


\  21.  The  sides  of  a  triangle  are  a,  x,  y,  subject  to  the  con- 
dition  X  +  y  =  m,  a   constant.      Prove   that    the   triangle   of 


maximum  area  is  isosceles,  and  that  x  =  y 


The  area  of 


a  triangle  in  terms  of  its  sides  =  A  =  Vs(s  —a)  (.s  —  x){s  —  y) 
in  which  s  =  ^  sum  of  the  sides.    By  condition,  2s=x-^y  +  a 


=  m  +  a,  whence  s ; 


Hence 


APPLTCATIOXS   OF    SUCCESSIVE    DIFITERENTIATION.      107 


A  =  ^  Vm-—  a^  Vet"  —  m-+  4  mx  —  4  x^, 

A'=  mx  —  or,  f{x)  =  m  —  2x  =  0,  and  x  =  —  ' 

7th 

From  x  +  y  =  m,  y=-' 

1/22.  Find   the   point   P  of   least   illumination   on   the   line 

joining  two  lights  A  and  B,  the  intensity  at  a  unit's  distance 

of  A  being  b,  and  that  of  B  being  c,  knowing  that  the  intensity 

varies  inversely  as  the  square  of  the  distance. 

If  the   distance  between  A  and  B  is  a,  and  AP=x,  the 

I)  c  ab 

illumination  at  P=  --\ — ;  whence  x  =  — • 

x-      {a  —xy  ji  _l_  gi 

23.  Kequired  the  height  of  a  light  directly  above  the  centre 
of  a  circle  whose  radius  is  i?  when  the  perimeter  is  most  illu- 
minated, knowing  that  the  illumination  varies  directly  as  the 
sine  of  the  angle  of  incidence,  and  inversely  as  the  square  of 
the  distance. 

Let  X  =  height. 

X  R 

Then  the  illumination  = ,    .-.  x  =  — -- 

{x'-R')'^  V2 

24.  The  base  of  a  prism  is  a  given  regular  polygon  whose 
tea  is  A  and  perimeter  P.     The  prism  is  surmounted  by  a 

regular  pyramid  whose  base  coincides  with  the  head  of  the 
prism.  Find  the  inclination  0  of  the  faces  of  the  pyramid  to 
the  axis,  in  order  that  the  whole  solid  may  have  a  given  volume 
C  with  the  least  possible  surface. 

Let  a;  =  height  of  prism.  Then  its  volume  is  Ax  and  sur- 
face Px. 

Let  a  —  perpendicular  from  centre  of  polygon  on  one  side. 

mi  Aa  cot  6  •  ,-,  T  J,  ,1  1  1  Pa  cosec  6 
Then  is  the  volume  oi  the  pyramid,  and 

its  surface.     By  condition 

.,      .a  cot  6\      ri                 ^  -~\  "^  ^^t  0 
ji[x -\ —  1  =  O,    . •.  a;  — <»— ~ • 


108  THE  DIFFERENTIAL   CALCULUS.  * 

The  surface  which  is  to  be  a  minimum  is  P(x-\ Y 

or,    substituting  the    value    of    x, ^  a  cot  6  +  ^a  cosec  0. 

Hence /'(d)  =  -cosec2^-"  cot  ^cosec^=0,  .-.  2 cosec  d= Scot  d, 
or  cos  d  =  f ,  and  0  =  cos~'  |. 

25.  Prove  that  the  minimum  tangent  which  can  be  drawn 

to  an  ellipse  is  divided  at  the  point  of  tangency  into  segments 

,    which  are  equal  to  the  semi-axes. 

a-  b^ 
If  (x,  y)  is  the  point  of  tangency,  — ,  —  are  the  intercepts  of 

X    y 

the  tangent,  and  the  length  of  the  tangent 


\  X-      y^       \x-      a-  —  XT 


a-  h^ 

Hence  the  function  to  be  examined  is  -7,  +  -^ 


a;-      a-  —  XT 


From  /'(.'r)  =  0  we  find  x^oyl— — ,  and  from  the  equa- 

\a-\-h 

tion  of  the  ellipse,  yz=h-J\ If  P  is  the  point  of  tan- 

gency,  and  T  the  point  where  the  tangent  meets  X, 


\"^\x        )      ^a  +  6^l,(„  +  j)j;      \    a  +  6 
and  in  like  manner  the  other  segment  may  be  shown  to  be  a. 

26.  In  the  straight  line  bisecting  the  angle  ^  of  a  triangle 
ABC,  a  point  P  is  taken.  Prove  that  the  difference  of  the 
angles  APB,  APC  is  a  maximum  when  AP 
is  a  mean  proportional  between  AB  and 
AC. 

Let  AC=  a,  AB  =  b,  PAB  =  m,  AP=  x. 

Draw  PE  and   PF  perpendicular  to  the 
sides.     Then  the  function  is  ^         ''•g-  23. 


APPLICATIONS    OF    SUCCESSIVE   DIFFERENTIATION.      109 


APC  -  APB  =  EPC  -  FPB  =  tan-i  ^  -  tan-^^ 

EP  FP 

^ih  —  xcosm , 


,        .a  —  X  cos  m 

—  fan— i 

X  sin  m 


=  tan" 


tan 


a.*  sm  m 


whence 

/'  (..)  = ^ ^- — 

a^  +  a^  —  2  ax  cos  m      ar'  +  6-  —  2  bx  cos  ^m 

from  which  we  find  x  =  -y/ab- 


=  0, 


27.  A  paraboloid  of  revolution  whose  axis  is  vertical  con- 
tains a  quantity  of  water  into  which  is  sunk  a  given  sphere, 
the  quantity  of  water  being  just  sufficient  to  cover  the  sphere. 
Find  the  form  of  the  paraboloid  such  that  the  quantity  of 
water  may  be  a  minimum,  knowing  the 
volume  of  the  paraboloid  to  be  one-half 
that  of  the  circumscribing  cylinder. 

Let  R  =  radius  of  sphere,  z  =  OH, 
the  height  of  the  water  when  the  sphere 
is  sunk,  and 


Fig.  24. 


2/2  =  Ix 


(1) 


be  the  equation  of  the  parabola,  in  which 
I  is  the  unknown  parameter.     The  equa-    Y 
tion  of  the  circle  is 

f-^(x-ocy=R', 

or,  since  OC  —  z  —  R, 

y-  +  {x-z-l-Ry=E\ 
Combining  (1)  and  (2),  we  have 
lx  +  (x-z+  Ry  =  R-, 


(2) 


whence        ,^2(z  -  R)-l  ±VAR^ +  1^ -^zl  +  ^Rl^ 

2 

But  the  circle  is  tangent  to  the  parabola,  and  there  can  be 
but  one  value  for  x,  and  hence 


110  THE   DIFFERENTIAL   CALCULUS. 

"'-~Ti (3) 

The  vol.  of  water  =  vol.  of  paraboloid  -  vol.  of  sphere 

since  (1)  gives  HS'  =  Iz.    Substituting  the  value  of  z  from  (3) 
the  function  becomes 

or  Y>-(i  +  2R)\ 

I 

whence /'(/)  =  0  gives  I  =  ?^  R,  which  determines  the  form  of 
the  paraboloid. 


CHAPTER   IV. 

FUNCTIONS   OF   TTVO    OR   MORE  VARIABLES. 

87.  A  partial  differential  of  a  function  of  two  or  more  vari- 
ables is  its  differential  on  the  hypothesis  that  only  one  of  the 
variables  changes.  Thus,  if  w  =  sin  x  log  y  +  zar,  and  x  only  is 
supposed  to  change,  y  and  z  being  regarded  constant,  the  par- 
tial differential  of  u  with  respect  to  x  is  cos.r  log  ydx  +  2 zxdx ; 
and  the  partial  differentials  of  u  with  respect  to  y  and  z  are 

dy  and  x'dz,  respectively. 

88.  Notation.  To  distinguish  the  partial  differentials,  the 
variable  with  respect  to  which  the  function  is  diiferentiated  is 
written  as  a  subscript,  thus  : 

d^u  —  (cos  X  log  y  +  2  zx)  dx,   d^u  — dy,   d^u  =  x^dz, 

which  are  read  '  the  a;- differential  of  w,'  etc. 

d  tt 

-J-  is  evidently  the  rate  of  the  function  so  far  as  its  rate 

depends  upon  the  rate  of  x. 

89.  A  partial  derivative,  or  partial  differential  coefficient,  is 

the  ratio  of  a  partial  differential  to  the  differential  of  the  variable 
lohicli  is  supposed  to  vary.  Thus,  in  the  above  case,  the  partial 
derivatives  of  u  with  respect  to  x,  y,  and  z,  respectively,  are 

du  ,  _         du      sin  a;     du        , 

-y-  =  cos  x  log  y  +  2zx,    -j-  = ?    -^  =xr, 

dx  °^  dy         y       dz 

the  subscripts  being  omitted  as  the  denominators  indicate  the 
variable  with  respect  to  which  the  differentiation  is  performed. 

Ill 


112  THE   DIFFERENTIAL   CALCULUS. 

A  partial  derivative  is  the  ratio  of  the  rate  of  the  function  to 
that  of  the  variable  supposed  to  vary  so  far  as  the  rate  of  the 
former  depends  upon  that  of  the  latter,  and  the  function  is  an 
increasing  or  a  decreasing  function  of  any  one  of  its  variables 
according  as  the  corresponding  partial  derivative  is  positive  or 
negative  (Art.  22). 

Since  the  first  derivative  is  the  factor  by  which  the  differen- 
tial of  the  variable  is  multiplied  to  obtain  the  differential  of 
the  function,  the  partial  differentials  are  also  represented  by 
the  notation 

clu  ,      dw  -       du  ^ 

-7-  clx,    -T-  ay,    -I-  az,   etc., 

dx      '    dy    •^'    cZz      '         ' 

which  are  equivalent  to  d^xi,  d^u,  d.,u,  etc. 

Examples.     Find  the  partial  differentials  of ; 

1.  u  =  (or'  +  f)^.      ^dx  =       ^'^^^      ,    ^-^dy  =       y^y      . 

^^^  {x^  +  f)^     "^y  {:>?  +  f)^ 

o  •   -\X  dii  J  dx  du  ,  xdy 

2.  u  =  sin    -•  —  dx  =  —  ,    —  dy= ^ 

y  dx  ^y^  —  x-    dy  y^y^-Qi? 

3.  u  =  y".  — dx  =  zy"  log  ydx,     —dy=^  xzy"~^dy, 

dx  dy 

— dz  =  xy"  log  ydz. 
dz 

Find  the  partial  derivatives  of  : 

4.  u  =  s\\\{xy).         ~—  —  ycos,{xy),    —  =  a;cos(a;y). 

5.  u  =  2/°'°*.  —  =  ?/^'"'=  log y  cos  X, 

dx 

du       ■          .:„__,       sin  a; 
—  =  sm X  •  v^'""  '  = 

dy  ^eoversx 

n  T      /     ,     \     da      du         1 

6.  «  =  log(aj4-?/)-    -r=-r  = 


dx      dy      x-\-y 


FUNCTIONS   OF    VARIABLES.  113 

90.  The  total  differential  of  a  function  is  its  differential  ob- 
tained on  the  hypothesis  that  all  its  variables  change. 

Since  the  total  differential  of  u  =f{x,  y,  z,  etc.)  can  contain 
only  the  first  powers  of  dx,  dy,  dz,  etc.,  it  will  be  of  the  form 
du=f{x,  y,  z,  QtQ.) dx+f_{x,  y,  z,  etc.)dy-^f{x,  y,  z,  etc.)d2  +  etc., 
in  which /i(x,  y,  z,  Qtc.),f{x,  y,  z,  etc.),  etc.,  represent  the  col- 
lected coefficients  of  dx,  dy,  etc.  But  if  all  the  variables  except 
X  be  regarded  constant,  dy  =  dz  =  etc.  =  0,  and  all  the  terms 
vanish  except  the  first,  which  is  the  partial  differential  with 
respect  to  x ;  and  if  all  the  variables  except  y  be  regarded  con- 
stant, all  the  terms  vanish  except  the  second,  which  is  the  par- 
tial differential  with  respect  to  y ;  and  so  on.  Thus  all  the 
terms  of  the  second  member  will  be  obtained  by  differentiating 
u  in  succession  as  if  all  the  variables  but  one  were  constant. 
Hence  the  total  differential  of  a  function  is  the  sum  of  its  partial 
differentials. 

Illustration.     Let  u  =  3  ax^y  +  zrV. 

du  =  d^u  +  dyU  +  d^u 

=  (6  axy  -\-  2rV)  dx  +  3  aardy  -f  3  z^e'dz.  (1) 

Equation  (1)  is  evidently  true  whether  the  variables  be 
dependent  or  independent.  If,  however,  the  variables  are 
dependent,  the  total  differential  may  be  expressed  in  terms  of 
any  one  of  them.     Thus  if 

y  =  bx,  2  =  sin  x,  whence  dy  =  bdx,  dz  =  cos  xdx, 

(1)  becomes 

du  =  (9  abay^  -f-  e''  sin"  x  —  3e'  sin^  x  cos  x)  dx ; 

or  the  same  result  might  be  obtained  by  substituting  the  values 
of  y  and  z  in  the  original  function,  giving 

u  =  3abx"  -f-  e"  sin^  x, 

whence,  differentiating,  we  have  (2),  as  before. 

If  Equation  (1)  be  divided  by  dt,  we  have  the  rate  of  the 


114  THE   DIFFERENTIAL   CALCULUS. 

function  in  terms  of  the  rates  of  its  variables ;  and  if  the  vari- 
ables are  dependent,  the  rate  of  the  function  may  be  expressed 
in  terms  of  that  of  any  one  of  its  variables  assumed  as  the 
independent  variable.  Thus,  dividing  (2)  by  dt,  we  have  the 
rate  of  u  in  terms  of  the  rate  of  x. 

Examples.     Find  the  total  diiferentials  of : 

1.  u  =  axry  +  by^'x.     du  (3  aa^y  +  by^)  dx  +  {ax^  +  3  by^x)  dy. 

o  i.     -i^  J        ydx  —  xdy 

2.  u  =  tan     -•  du  =  ^ •-  • 

y  ar-f-?/2 

ydx 

3.  u  =  log  x".  dit  = H  log  xdy. 

4.  u  =  r  cos  6.  du  =  cos  6dr  —  r  sin  Odr. 

xii  ,        a?dy  4-  y-dx 

o.  u  =  — '^ —  da  =  —       ^ — 

x  +  y  {x-i-yy 

6.  ?t  =  e'a".         '        du  =  cO'e'dx  +  e^a"  log  ady. 

7.  u  =  tan~^  (xy) .       du  =  ^ — — — -^• 

8.  w  =  x^.  du  =  y3^~^dx  +  a;"  log  xdy. 

9.  ?6  =  a''  +  e~*'2;  +  sin  v. 

du  =  a"'  log  adx  —  ze^Hly  +  e~''d2:  -f-cos  udy. 

,x  —  y       ,        ydic  —  a;d?y 
10.  «  =  tan-i — ~-     da  =  '^—;r-. — ^^ 
x-{-y  X'  +  y^ 

91.   The  total  derivative,  or  total  differential  coefficient,  of  a 

function  is  the  ratio  of  the  total  differential  to  the  differential  of 
the  independent  variable. 

Thus,  if  M  =f{x,  y,  z),  we  have  for  the  total  differential,  by 
Art.  90, 

du  ,        du  ,        du  , 
^^  =  d^^'^'  +  d2,^^  +  d^^"5  (1) 


FUNCTIONS    OF    VARIABLES.  115 


and  if  x  be  the  independent  variable, 


rdtt"|_ 
Idxj- 


du      du  dy      du  dz 

1 ^H (2) 

dx      dy  dx      dz  dx  ^  ' 

The  student  will  observe  that  the  du's  are  not  the  same  in  the 
above  formulae.  In  the  first  member  of  (1)  d?t  is  the  total 
dili'erential  of  u,  while  in  the  second  member  du  is  a  partial 
differential,  the  notation  serving  to  distinguish  the  partial 
differentials  from  each  other  and  from  the  total  differential. 

To  cancel  the  equal  factors  from  the  terms  of  the  second 
member  would  be  to  destroy  the  means  of  distinguishing  the 
dw's,  no  two  of  which  are  the  same.     In  fact,  (1)  is 

du  =  d^u  -f-  d^n  +  d^xi. 

In  forming  (2)  from  (1)  the  first  member  becomes  a  total 
derivative,  the  bracket  being  used  to  distinguish  it  from  the 

partial  derivative  —  in  the  second  member.     It  is  further  to 
dx 

be  observed  that  while  (1)  is  true  whether  the  variables  be 
dependent  or  independent,  (2)  has  no  significance  unless  the 

(Jv  dz 
variables  are  dependent ;  for  -~,  — ,  cannot  be  evaluated  unless 

CttV     CtJb 

y  =  cf>{x),z  =  ip{x). 

The  total  derivative  is  evidently  the  ratio  of  the  rate  of  the 
function,  on  the  hypothesis  that  all  its  variables  change  to  the 
rate  of  the  independent  variable. 

92.  The  total  derivative  with  respect  to  any  independent 
variable  may  be  formed  in  like  manner  by  dividing  the  total 
differential,  which  is  always  the  sum  of  the  partial  differen- 
tials, by  the  differential  of  the  independent  variable,  under- 
standing that  the  independent  variable  is  connected  with  the 
others  by  auxiliary  relations. 

Thus,  given  w  =f(x,  y,  z)  and  x  =  <f>i(tv),y  =<f>o(w),  z  =  </>3(w), 
to  form  the  total  derivative  with  respect  to  w,  we  have 
du  ,       du  ,       du  , 


116 


THE   DIFFERENTIAL   CALCULUS. 


whence 


Fdul 
\_dwj 


du  dx      du  dy      du  dz 
dx  dw      dy  dw      dz  dtv 


111  this  case  the  bracket  is  not  necessary,  but  it  is  usual  to 
enclose  the  total  derivative  in  brackets  whatever  the  indepen- 
dent variable. 


Examples.      1.    Given   u  =  2axy  -^logx,   and   a;  =  siny,  to 
form  the  total  derivative  of  u  with  respect  to  y. 

du  ^        du  ^ 
du   =  ^r-dx  -\--r-  dy, 


du 
_dy_ 


dx 

du  dx     du 

dx  dy      dy 


From  the  given  function  Ave  find  the  partial  derivatives 


du 
dx 
dx 


1     ^" 


and  from  x  =  sin  y,  —  =  cos  y.     Substituting  these  values, 


du 
dy 


=  (2  ay  -{-  -j  cos  y  -\-2ax 
=  2a(ycosy  -{•  smy)-\-  cot?/. 


The  same  result  would  be  obtained  by  first  substituting  the 
value  of  X  in  the  function  and  then  differentiating. 

2.  u  —  y--}-z*-i-zy,    y  =  smx,    z  =  cosx. 


du 
dx 


=  cos2a/'(l  —  sin2a;). 


3.  M 


1  ^  —  V 
tan  '■ — r-^,   x  =  e'',   y  =  e^ 


x  +  y' 

du 
dz 


2e-' 

By  substituting  the  values  of  x  and  y  in  u  and  then  dif- 
ferentiating, the  student  may  compare  the  two  processes.  In 
this  case  the  use  of  the  formula  is  more  expeditious. 


FUNCTIONS   OF   VARIABLES. 


117 


4.  It  =  tan  ^-,   xr-\-y-=  R-. 

y 

fdul  _      1 
(hf    ~      X 

.-  •    2;  „ 

o.  M  =  Sin  -,    2  =  6*,   y  =  XT. 


[du~\  e'       e 


6.  u  =  yz,  y  =  e',  z 


4ar^+12ar'-24a;  +  24. 


rdH~\ 
\dx\ 


ex" 


93.  Implieit  function  of  two  variables. 

Let  f{x,  y)  =  0.     Representing  the  function  by  u  we  have 

u=f{x,  y)  =  0. 

Since  the  only  possible  values  of  the  variables  are  those 
which  render  the  function  zero,  u  is  constant,  and  hence  its 
differential  is  zero.     Therefore 

,        du  J     ,  da,       rt 

du  =  —  dx-\ dy  =  0, 

dx  dy 


whence 


du 

dy_ 

dx 

dx 

~d^' 

dy 

(1) 


or  the  first  derivative  of  an  implicit  function  is  the  negative  ratio 
of  its  partial  derivatives. 

The  above  depends  solely  upon  the  fact  that  du  is  constant ; 
hence  (1)  is  true  when  f{x,  y)  =  a,  where  a  is  any  constant. 
dy 
dx 


Thus,  f  -  2px  =  0, 


-2^=^-     Again,  i^+a;^=. 


=  —  —  =—-.     These  results  might  of  course  be  ob- 
dx  2y  y 

tained  directly  by  the  ordinary  process  of  differentiation,  but 
it  is  often  useful  to  employ  the  value  of  the  derivative  in 
terms  of  the  partial  derivatives  as  given  in  (1). 


118  THE   DIFFERENTIAL   CALCULUS. 

Examples,     Form  by  the  above  method  the  derivatives  of : 

1.  u  =  3  ascry  —  2  ay-x  =  c. 

6  axy  —2aff_y{2y  —  6  x) 
'3ax'  —  4:ayx  ~  x('dx  —  Ay) 

o  1  1  n  ^y     yx\ogy-y^ 

2.  u  =  x  losf  y  —  y  losr  a;  =  0.  :j~  =  — t :3* 

o  J      ^     ty  dx     yx  log  x  —  ar 


du 

dy 
dx~ 

dx 
du 

dy 

3.  M  =  y  -f  ar  —  3  mxy  =  0. 

4.  II  =  ?/e"*  —  a-r™  =  0, 

5.  u  =  sin  (x?/)  +  tan  (.ry)  =  a. 

6.  M  =  o?/''  —  x^y  —  ax^  =  0. 


dy  _  my  —  d? 
dx  y^  —  mx 
dy  my 


dx     x{l  +  ny) 

^=  -I. 
dx         X 

dy  _  3  a^?/  +  3  ax^ 
dx        3  ay^  —  x" 


94.   Evaluation  of  the  first  derivative  of  an  implicit  function. 

Let  f{x,  y)  =  0,  in  which  x  is  the  equicrescent  variable. 

du 

Then  dy^_dx 

dx         du 

dy 

may  be  a  function  of  both  x  and  y  and  assume  the  illusory 

form  -  for  particular  values  of  x  and  y.     In  such  a  case  we 

may  eliminate  y  from  -^  by  means  of  f{x,  y)  =  0,  and  then 

proceed  as  in  Art.  74 ;  or,  since  y  is  a  function  of  x,  we  may 
apply  the  process  of  Art.  74  directly,  without  eliminating  y, 
forming  the  successive  derivatives  of  the  numerator  and 
denominator  with  respect  to  x  until  a  pair  is  found  whose 

ratio  does  not  become  -  for  the  particular  values  of  the  variar 

bles.     Thus,  if  u  =  y^  —  xry  —  x*  =  0, 


FUNCTIONS   OF   VARIABLES. 


119 


dy 
dx 


du 

dx  _  '2xy  -{•■ia? 

du~  3y^  —  x^ 

dy 

0 


which  for  x  =  y  =  i)  becomes  -.     By  Art.  74, 


dy  _2xy  4-4ar^' 
dx~   3y^  —  X- 


2y +  2.^  +  12^ 


«^|-2^ 


0,0 


^dy      ^    d'y      ^dy 
dx  dx-        dx 


.dy' 


G-^  +  ef^-2 


d^ 
'd^ 


4^ 
dx 


0,0 


,dl 

dx- 


,df     ^dy      .dy 


dy 


Hence  6-^  —  2^^  =  4^,  or  V^  =  0,  and  ±  1. 
dx^        dx        dx         dx 


Having  seen  that  the  true  value  of  an  expression  which  assumes  the 

form  -  for  a  particular  value  of  the  variable  is  its  limit  (Art.  74),  it  vi^ould 

seem,  since  a  quantity  can  have  but  one  limit  (Art.  64),  that  -^  in  the 

dx 

above  example  could  have  but  one  value.     That  it  may  have  several  values 

will  appear  in  Art.  114. 


dy 
Examples.    1.  If  u  =  x*-{-2axhj—  ay'^=  0,  shoAv  that  ~  =  0, 

or  ±  V2  when  x  —  y  =  0. 


dy  _      4  .r^  -f  4  axy 
dx~      2  aaf  —  3  ay' 


12  a:?  +  4  ay  +  4  ax 


dy 
dx 


0,0 


bay-j Aax 


CM      .   4    dti      ,    dij      ,      d'y 

24a;  +  4a  /  +  4a  ,-  +  4aa;-T4, 
dx  dx  dx- 


dy' 


6a^-f  6a2/^-4a 


'da^ 


0,0 

^dy 
dx 


Jo,o 


<:-^« 


w»-l(S-)=»- 


120  THE   DIFFERENTIAL   CALCULUS. 

2.  If  u  =  X*  -  af  +  2  axf  +  3  ax'y  =  0,  show  that  ^  =  0,  3, 
or  —  1,  when  x  =  y  =  0. 

3.  u  =  x^  -  a?xy  +  f  =  0.      Prove  that  g  =  0,  or  a%  when 
x  =  y  =  0. 

4.  u  =  X'  +  ax^y  -  af  =  0.     Prove  that  ^  =  0,  or  ±  1,  when 

X=y=:0. 

5.  u  =  ahf  -  2  abxhj  -  .r'  =  0.      Prove  that   5^  =  ±  0,  when 

,,  ax  ' 

x  =  y  =  (). 


CHAPTER   V 


PLANE     CURVES. 


CURVATURE. 


95.  A  curve  is  concave  npivard  at  any  of  its  points  when  its 
tangent  at  that  point  lies  below  the  curve,  arid  is  convex  ujncard 
when  its  tangent  lies  above  the  curve. 

When  a  curve  is  concave  upward,  its 
slope  increases  with  cc;  hence  if  y=zf{x) 

be  its  equation,  f\x)-\--^  is  an  increas- 
ing function.  But  the  first  derivative  of 
an  increasing  function  is  positive,  and  the 

first  derivative  of /'(a;)   is  f"{'x)  =  -~- 

Hence /"(a;)  is  positive  tvhen  the  curve  is 
concave  upivard. 

If  the  curve  is  convex  upward,  its  slope 
decreases  as  x  increases ;  f'(x)  is  a  decreas- 
ing function,  and  its  derivative  is  therefore 
negative.      Hence  f"{x)  is  negative  when  the  curve  is  convex 
upivard. 


96.  A  point  at  which,  as  x  increases,  the  curvature  changes 
from  concave  to  convex  upward,  or  vice  versa,  is  called  a  point 
of  inflexion.  At  a  point  of  inflexion  the  tangent  evidently  cuts 
the  curve. 

Since  on  one  side  the  curve  is  convex  and  on  the  other  con- 
cave upward,  the  analytic  condition  for  a  point  of  inflexion  is 

121 


Fig.  27. 


X 


122  THE   DIFFERENTIAL   CALCULUS. 

(Art.  95)  a  change  of  sign  mf"{x).  Hence  all  values  of  x  cor- 
responding to  such  points  are  roots  of  the  equations /"(cc)  =  0, 
f"{x)  =  cc.  These  roots  are  critical  val- 
ues, and  do  not  correspond  to  points  of 
inflexion  iiidess  accompanied  by  a  change 
of  sign  in  f"(x). 

In  approaching   a   point    of    inflexion 
f'{x)   is  increasing   (or  decreasing),    and  after  passing  this 
point  is  decreasing   (or  increasing) ;  hence  /'  (x)   is  either  a 
maximum  or  a  minimum  at  a  point  of  inflexion. 

Examples.  Examine  the  following  curves  for  curvature 
and  points  of  inflexion : 

1.  a;  =  log  y,  the  logarithmic  curve. 

f"(^x)  =  y,  which  is  always  positive,  since  negative  numbers 
have  no  logarithms.  The  curve  is  therefore  always  concave 
upward  and  has  no  point  of  inflexion. 

2.  y'-+  x^=  R%  the  circle. 

7?- 

f"(x)= -,  which   is   negative  when   y  is  positive,  and 

y3 

positive  when  y  is  negative ;  hence  the  curve  is  convex  upward 
above,  and  concave  upward  below,  X.  f"(x)  has  two  signs, 
but  does  not  change  sign  for  increasing  values  of  x,  and  there 
is  no  point  of  inflexion. 

o.  xy  =  m,  the  hyperbola. 

2  m 
/"(a;)  =  — ^,  which  has  the  sign  of  x.  The  curve  is  there- 
fore concave  upward  in  the  first,  and  convex  upward  in  the 
third,  angle.  f"(x)  changes  sign  at  a;  =  0;  but  when  a;  =  0, 
y  =  00,  the  curve  being  discontinuous,  and  there  is  no  point  of 
inflexion. 

4.  cry  =  4a^(2a  —  y),  the  witch. 

f"(x)  =  2y  A?!nA^ .     Points  of  inflexion  at  a;  =  ±  —  • 
(.r-+4a-)-  V3 


CURVATURE.  123 

5.  ay-z=oc?,  the  semi-cubical  parabola. 

G.  y  =  sin  x,  tlie  sinusoid. 

7.  X  =  lo^y.     A  point  of  inflexion  at  x  =  S,  where  the  curva- 
ture changes  from  convex  to  concave  upward. 

8.  y^  =  arx,  the  cubical  parabola. 

2  a* 

y "  (.1-)  =  — ^:^,  which  is  positive  when   ?/  is  negative,  and 

y 

negative  when  y  is  positive ;  hence  the  curve  is  concave  up- 
Avard  in  the  third,  and  convex  upward  in  the  first,  angle. 
f"(x)  changes  sign  at  ?/=0,  whence  x  =  0,  passing  through 
infinity,  and  the  origin  is  a  point  of  inflexion. 

9.  y(a*-b')  =  x{x-ay-xh\ 

A  point  of  inflexion  when  x  =  la. 

10.  y  = 


d'-\-x^ 
11.  y  =  tan  X. 

97.  Kate  of  curvature.  A  plane  curve  may  be  defined  as 
the  locus  of  a  point  which  always  moves  along  a  straight  line 
while  the  line  always  turns  around  the  point. 

Since  the  direction  of  motion  is  always  that  of  the  line,  the 
line  is  the  tangent  to  the  curve.  Were  the  line  to  remain  fixed, 
the  locus  would  be  a  straight  line,  that  is,  if  the  tangent  does 
not  turn  about  the  moving  point  there  is  no  curvature ;  hence, 
if  <^  be  the  angle  which  the  tangent  makes  with  any  fixed  line 
as  X,  the  curvature  will  depend  upon  the  change  of  ^. 

Since  in  the  circle  equal  arcs  subtend  equal  angles  at  the 
centre,  the  normal,  and  therefore  the  tangent,  turns  through  the 
same  angle  for  every  unit  of  path  describetl  by  the  generating 
point,  and  the  curvature  of  the  circle  is  therefore  constant 
whatever  the  unit  by  which  it  is  measured. 


124  THE   DIFFERENTIAL   CALCULUS. 

It  is  evident  that  if,  in  passing  a  second  time  through  any 
point  of  a  given  curve,  the  velocity  of  the  generating  point  be 
m  times  what  it  was  before,  the  rate  of  turning  of  the  tangent 
at  that  point  will  also  be  m  times  its  former  rate ;  or  that  the 
ratio  of  the  rate  of  turning  of  the  tangent  to  the  velocity  in 
the  curve  is  constant.     Hence 

dcfi 

di  _(l<i> 

ds  ~  ds 

dt 
is  a  constant  for  the  same  point,  whatever  the  velocity.  This 
expression  is  evidently  the  rate  of  turning  of  the  tangent  per 
unit  of  length  of  the  curve,  and  may  be  taken  as  a  measure  of 
the  curvature.  This  measure  is  independent  of  t,  as  it  should 
be,  for  the  curvature  is  a  geometric  proj)erty  of  the  curve 
independent  of  the  time  of  its  description. 

dd> 
Since  the  rate  -j-  is  the  amount  by  which  <^  would  change 

for  a  unit's  length  of  path,  were  its  rate  to  remain  through  this 
distance  what  it  was  at  its  beginning,  the  curvature  at  any 
point  of  a  plane  curve  is  that  of  a  circle  which  has  a  common 
tangent  with  the  curve  at  the  point  considered.  This  circle  is 
called  the  circle  of  curvature,  and  its  radius  the  radius  of  cur- 
vature. 

98.    To  express  — ^  in  terms  of  the  coordinates  of  the  generating 
ds  ' 

point. 

From  Art.  25  we  have 


ds=-y/da?-\-dy'^, 
and,  reckoning  <^  from  the  axis  of  X, 

tan  <^  =  — ",  whence  sec^  <f>d(f>  = — —, 
dx  dx 

dry  d^y  d'-y 

dx  dx  dx 

or  o^  = 


sec^<^      1  +  tan''<^      -,  ..dy'' 
dx'^ 


CURVATURE.  125 

d-y                           ^y 
Hence  -^  =  7 T^^ =7 TTTT*         i^) 

To  find  therefore  the  curvature  of  a  plane  curve  y=f{x), 
differentiate  its  equation  twice  and  substitute  in  (1)  the  values 
of  the  first  and  second  derivatives. 

99.  Curvature  of  the  circle. 

in  o  ,     .,      ry,  dy         X  X  d-y  R      ,  .  , 

1^  rom    X--\-  y-=  li-,_±  = = -::::::  ;    — ^  = -,  which 

f^x         y         Vi2--a^    ^^         2/ 
will  be  ±  as  ?/  is  :f .     Hence 

m 

dct>  _        f        _       1 
ds       A      a^\|  R' 

or  the  curvature  of  a  circle  is  the  reciprocal  of  its  radius. 

CoR.  1.     Since  —  =1  when  i?=  1,  the  unit  of  curvature  is 
ds 

seen  to  be  the  curvature  of  the  circle  whose  radius  is  unity.- 

Cor.  2.  The  curvatures  of  any  two  circles  are  inversely  as 
their  radii. 

100.  Radius  of  curvature.  Since  the  curvature  of  any  plane 
curve  at  a  given  point  is  that  of  its  circle  of  curvature  at  that 
point,  and  the  curvature  of  this  circle  is  measured  by  the  re- 
ciprocal of  its  radius,  we  have,  if  p  be  this  radius, 

p      ds^ 
or  p='^'      ^        ^"^ 


dx" 


126     y  THE  DIFFERENTIAL  CALCULUS. 

If  we  take  the  positive  value  of  the  radical,  the  radius  of 
curvature  will  be  ±  as  — ^ 


d^ 


is  ±  ;   that  is,  according  as  the 


curve  is  concave  upward  or  downward  at  the  point  considered. 
The  sign  of  p  may  thus  serve  to  determine  the  direction  of 
curvature. 

CoR.  1.    Since  — ^  =  0  at  a  point  of  inflexion,  the  radius  of 
dar 

curvature  at  a  point  of  inflexion  is  infinite,  and  the  curvature 
zero. 

CoR.  2.  Since  the  circle  of  curvature  at  any  point  has  a 
tangent  in  common  in  the  curve,  the  radius  of  curvature  is  a 
normal  to  the  curve. 


101.   Coordinates  of  the  centre  of  curvature. 

Let  C  be  the  centre  of  curvature  of  the 
curve  MN  at  any  point  P,  and  a,  /?  the  co-    i' 
ordinates  of  C     Then 


a  =  OD=OB-DB=x 

dy 


/t)  sin  <j!> 


(3  =  DC  =  BP  -\-  SC  =  y  +  peostji 
dx 


ds 


Substituting  the  values  of 


ds  =  -Vdx-  +  dy-  and   p  =  — 
\ 


\dxj 


dry 


a  —  X  —  - 


dx 


dx^ 


dx  \dx 


daP 


(1) 


CURVATURE. 


127 


102.  Maximum  or  minimum  curvature.   Since  the  curvature 

is  measured  by  ->  it  will  be  a  maximum  or  minimum  when  p 

is  a  minimum  or  maximum.  It  is  further  evident  that  if  a 
curve  is  symmetrical  with  reference  to  the  normal  in  the  vicin- 
ity of  the  point  of  contact,  the  curvature,  if  not  constant,  will 
be  a  maximum  or  a  minimum  at  that  point. 

103.  In  the  vicinity  of  a  point  of  maximum  or  minimum  cur- 
vature, the  circle  of  curvature  lies  wholly  on  one  side  of  the  curve  ; 
at  all  other  points  it  intersects  the  curve.  For  at  a  point  of  maxi- 
mum curvature  the  rate  of  turning  of  the  tangent  is  greater 
than  immediately  before  or  after,  while  the  rate  of  turning  of 
the  tangent  to  the  circle  of  curvature  remains  constantly  what 
it  was  at  the  point  of  contact ;  hence  the  circle  lies  within  the 
curve  at  this  point.  For  a  like  reason,  at  a  point  of  minimum 
curvature,  the  circle  lies  without  the  curve.  At  all  other 
points  of  the  locus  (except  when  it  is  a  straight  line)  its  cur- 
vature is  continually  increasing  (or  decreasing)  while  that  of 
the  circle  remains  the  same ;  on  one  side,  therefore,  the  curva- 
ture is  less  and  on  the  other  greater  than  that  of  the  circle, 
and  hence  the  curve  crosses  the  circle. 

Thus,  the  circle  of  curvature  lies  without  the  ellipse  at  the 
extremities  of  the  conjugate  axis,  within  at  the  extremities  of 
the  transverse  axis,  and  at  all  other  points  cuts  the  ellipse. 


Examples.     1.  The  parabola. 

_  „     ^        dy     p    cPy 

From  y^  =  2px,  -/=-»    -A  = 
^         ^       dx     y     dx^ 


Hence 


pi 


P  = 


1 +(&"'■ 

dx 


if+p')^ 


cPy 
daf 

1  + 


dy 
dx 


d^ 
djr 


=  Zx+p>, 


128  THE  DIFFERENTIAL   CALCULUS. 

At  the  vertex  y  =  x  =  0,  p=p,  a  =  p,  ^  =  0,  or  the  radius  of 
curvature  is  one-half  tlie  parameter  and  the  centre  of  curvature 
on  the  axis  twice  as  far  from  the  vertex  as  the  focus  is.  We 
observe  also  that  p  is  least  when  y=0,  or  the  curvature  at 
the  vertex  is  a  maximum. 

2.  The  ellipse. 

From  ay  +  6^x^=a^&S  ^  =  _  ?>^,  ^  =  _  _&!.. 
dx         a'y    da?         ci^if 

Hence  p  =  (^V  +  ^-^. 

At  the  extremities  of   the   conjugate   axis,  a;  =  0,  y  =  ±b, 
a? 

At  the  extremities  of  the  transverse  axis,  y  =  0,  x=  ±a, 
&2 

If  a=h  =  R,  p  =  Ri  the  radius  of  the  circle. 

3.  The  cycloid. 


From  :r=rvers-'^-V2^^^"=:?,  dy ^sj^ry-f^  ^=_I. 
r  dx  y  da?        y^ 

Hence  p  =  2-\/2ry,  or  the  radius  of  curvature  is  twice  the 
corresponding  normal  (Art.  48,  Ex.  9). 

At  the  highest  point,  y  =  2r,  p  =  4?';  at  the  vertex,  y  =  0, 
p  =  0. 

4.  The  logarithmic  curve. 

From  x  =  log^,  -^  =  ^,  — ^  =  -^ ;   hence  p  =  v^^  'rV  )  . 
dx     m   dar      mr  my 

If  a  =  e,  p  =  ^^ —    ^  -^  ,  and  if  a;  =  0,  whence  y  =  1,  p  =  2 V2, 

y 

the  radius  of  curvature  of  the  Naperian  curve  at  the  point 
where  it  crosses  Y. 


EVOLUTES  AND  ENVELOPES.  129 


5.  The  hypocycloid. 


From.  x^+y^  =  a^,  -^  =  —  -J~,  — ^  =  --i-^;  hence  p  =  3^/axy. 
When  either  x  or  y  is  zero,  p  =  0. 


6.  The  cubical  parabola. 
From  If  =  a-x,  p  =  -^^-^^ — - — ^• 

7.  The  semi-cubical  parabola. 

17  2        3  (4a  +  9a;)^   l 

From  aif  =  ar,  p  =  -^ ^x^. 

6a 

8.  The  catenary. 

From  2/  =  ^(e«  +  e"«),  p=-|'- 

9.  The  cissoid. 

i  rom  ?/  = ,  p  =  —5^ ^ ,  which  is  zero  when 

2a  — X  'S(2a  —  xy 

x  =  0,  and  infinity  when  x  =  2  a. 

EVOLUTES   AND  ENVELOPES. 

104.  The  locus  of  the  centre  of  curvature  of  a  given  curve  is 
called  the  evolute  of  the  curve. 

TJie  given  curve  is  called  the  involute. 

105.  Equation  of  the  evolute. 

Let  yz=f{x)  be  the  equation  of  the  involute.     The  coordi- 
nates of  its  centre  of  curvature  are  (Art.  101), 


a  =  X 


1  + 


\dxj 


dx       ^  \dx 


P  =  y  + 


d?y  '     ^      ^   ■         ^ 

da?  da? 

By  substituting  in  these  the  values  of  the  derivatives  ob- 
tained from  y  =  f{x) ,  we  obtain  the  values  of  a  and  /8  in  terms 


130 


THE   DIFFERENTIAL   CALCULUS. 


of  X  and  y.  Eliminating  x  and  y  between  these  results  and 
y  =f{x),  the  resulting  equation  between  a  and  /8  will  be  the 
equation  of  the  evolute. 

Examples.     1.   Eind  the  equation   of  the  evolute   of  the 
parabola. 

From  Art.  103,  Ex.  1,  Ave  have 

a='6x+p,     I3  =  -K 


Avhence 


a—p 
~3~"' 


y  =  —  ^-ip^. 


Substituting  these  values  of  x  and  y  in 
7/2=  2px,  we  have 

The  form  of  the  evolute  is  shown  in  the  figure. 
2.  Find  the  equation  of  the  evolute  to  the  ellipse 


whence  x= 


y  =  - 


b* 
b'l3 


Substituting   these    in  ah/-\-  b^x^=  a-b', 
we  find 

and  the  form  of  the  curve  is  shown  in  the  figure. 

3.  The  evolute  of  the  cycloid  is  an  equal  cycloid. 

Froma;  =  rvers-i-^-V2ry-2/^,  ^^^^ry-f^  ^  =  _21. 
r  dx  y  dx^         y^ 

Hence  x  =  a-  2  V-  2ry8  -  p\  2/  =  —  /8. 

Substituting  these  in  the  equation  of  the  cycloid, 


a  =  r  vers-'/'-  ^V  V-  2r;8  -  ^. 


(1) 


EVOLUTES    AND   ENVELOPES. 
If  the  given  cycloid  be  referred  to  the  axes  XiOiY^, 


131 


0,N  =x  =  CD  +  QP  =  MP  -{-QP=  MP  +  VMQ  •  QD 


which  is  of  the  same  form  as  (1).     Hence  the  evokite  is  an 
equal  cycloid,  0  being  its  highest  point. 

4.  Show  that  the  evolute  of  a  circle  is  a  point,  the  centre  of 
the  circle. 

The  usefulness  of  the  above  method  of  finding  the  equation 
of  the  evolute  is  limited  by  the  difficulties  of  elimination, 
although  the  method  is  general. 

5.  To  find  the  evolute  of  the  hypocycloid. 

dy  _      y^   d^y  _  1    a' 


From  x^  -\-y^  =  a^,  we  have 


dx 


,.J    dx^     3  »i_™t 


Hence  a  =  x  +  3  x^y^,  I3  =  y  -\-  oy^x^. 
To  eliminate  x  and  y  we  proceed  as  follows  : 

a+&=x  +  3  x^y^  +  3y*r^  +  y  =  (a:^  +  fy ; 
hence  (a+ B)^  —  x^  +  i/^. 

Similarly,      (a  — $)*  =  x^  —  y^. 
Hence,  (a+ /S)*  +  (a-)3)*=:  2.r*, 


yx' 


132  THE   DIFFERENTIAL   CALCULUS. 

and  [(a  +  ;8)^  +  (a  -  3)^]2  +  [(a  +  &)^  -  (a -  0)^^ 

6.  If  C  is  the  centre  of  an  ellipse,  CG  the  X-intercept  of  the 
normal  at  P,  and  0  the  centre  of  curvature  corresponding  to 
P,  prove  that  the  area  of  the  triangle  COG  is  a  maximum  when 
the  distance  of  P  from  the  conjugate  axis  is  one-fourth  the 
transverse  axis. 

106.  Envelopes.  The  equation  of  a  locus  is  a  relation  be- 
tween X,  y,  and  one  or  more  constants,  upon  which  latter  the 
position  or  form  of  the  locus  depends.  Thus,  the  constants  m 
and  b  fix  the  position  of  the  straight  line  y  =  mx  -f-  b ;  the  con- 
stants a  and  b  determine  the  form  of  the  ellipse  a^y--\-b-af=a^b^ ; 
while  the  constants  of  the  general  equation  of  a  conic  deter- 
mine both  its  position  and  form. 

The  constants  in  y  =f{x)  are  called  parameters. 

It  follows  that  if  different  values  be  assigned  to  one  of  the 
parameters  of  the  equation  y  =f(x),  the  resulting  eqiiations 
will  represent  a  series  or  system  of  curves  differing  from  each 
other  in  form,  or  position,  or  both.  Thus,  {x  —  my  +  y"^  =  R"^ 
is  the  equation  of  a  circle  whose  centre  is  on  X,  and  if  different 
values  be  assigned  to  m,  we  shall  obtain  a  series  of  equal  circles 
whose  centres  are  on  X. 

The  curve  which  is  tangent  to  all  curves  of  the  system  obtained 
by  the  coyitinuous  variation  of  any  one  of  the  parameters  in 
y  =  f{x)  is  called  the  envelope  of  the  system. 

The  constant  thus  supposed  to  vary  is  called  the  variable 
parameter. 

Thus,  in  the  case  of  the  above  circle  {x  —  my  +  y^  =  R^,  m 
being  the  variable  parameter,  the  envelope  of  the  system  is 
evidently  the  two  tangents  to  the  circle,  in  any  of  its  positions, 
which  lie  parallel  to  X. 


E VOLUTES   AND   ENVELOPES. 


133 


Denoting  the  variable  parameter  by  a,  the  general  equation 
of  the  system  may  be  represented  by 

f{x,  y,  a)  =  0. 


Fig.  32. 


107.   Equation  of  the  envelope. 

Let  SB  be  the  envelope  of  any  system  of  curves,  and  Q  the 
point  at  which  the  envelope  is  tangent  to  any  one  curve  of 
the  system  MN.     Let 

u=f{x,y,a)  =  0  (1) 

be  the  general  equation  of  the  system,  a  being 
the  variable  parameter,  and  P,  {x,  y),  any 
point  on  MN. 

Were  jTfiV  fixed,  that  is,  a  constant,  the  direc- 
tion of  P's  motion  would  be  determined  by 

du 
dy  dx 
dx~      du' 

dy 

But,  if  MN  is  not  fixed,  a  is  variable,  and 

,         du  ,     ,  du  7     ,  dii  ^        r> 
du  = -^  dx -\- -;-  dy  -\ da  =0, 


dx 


dy 


da 


whence 


du      du  da 
dy  dx      da  dx 

dx  du 

dy 


Now  when  P  coincides  with  Q,  these  values  of  -^  are  equal, 

dx 

since  MN  and  SR  have  at  Q  a  common  tangent.     Hence  at 

Q  —  —  =  0,  which  will  be  satisfied  if  da  =  0,  that  is,  if  the 
da  dx 

partial  derivative 


du  _  ,. 
da 


(2) 


134 


THE  DIFFERENTIAL   CALCULUS. 


The  coordinates  of  any  point  Q  of  the  envelope  must  there- 
fore satisfy  (1)  and  (2).  Hence,  to  determine  the  equation  of 
the  envelope  of  any  system,  combine  the  general  equation  of  the 
system  with  the  eqtiation  formed  by  jilacing  the  partial  derivative 
ivith  respect  to  the  variable  parameter  equal  to  zero^  eliminating 
the  parameter. 

Examples.  1.  Find  the  envelope  of  {x  —  m)--\-y-=R-,  m 
being  the  variable  parameter. 

XI  =  (x  —  my+  y-—  E^=  0,  —  =  —  2  (ic  —  m)  =0,  or  x  =  m. 

dm 

Substituting  this  value  of  m  in  {x  —  my+y-—  B^=  0  we  have 
y=±B,  two  straight  lines  parallel  to  X 

2.  Find  the  envelope  of  the  hypothenuse  of  a  right-angled 
triangle  of  constant  area. 

Let  OAB  =  c  be  the  constant  area,  and 
OA  —  a.     Then,  since 


OB-  OA 


c,     0B  = 


2c 


or 


+  y 


2( 


=  0, 


(1) 


Hence  the  equation  of  ^5  is 

a^'2c      ^' 
a 
_  2  ex 

a-  a 

in  which  a  is  the  variable  parameter.     Hence 

die 4  ex   ,  2  c 

da  a°       a' 

whence   a  =  2x.      Substituting  this   value   in    (1),   we   have 
xy  =  -,  the  equation  of  an  hyperbola  referred  to  its  asymptotes. 

3.  Find  the  envelope  of  an  ellipse  whose  eccentricity  so 
varies  that  its  area  remains  constant ;  knowing  the  area  of  an 
ellipse  to  be  irab. 


+  ^=0, 


E VOLUTES   AND   E^'VELOPES.  135 

We  have  -n-ab  =  m,  a  constant,  whence  a6  =  —  =  c,  a  constant. 

TT 

As  a  and  6  are  both  variable,  we  eliminate  either  parameter, 
as  b,  from  a-i/^+6V=a^6-  by  means  of  the  condition  ab=c,  and 

thus  obtain  u = a*  if  -\-  crxr — o?(? = 0 ;  whence  —  =4  ay — 2  ac^ = 0, 

or  a^=;^ — ■,,  which  in  a^  +  c^a^— aV=  0,  gives  xy=y-     Since 
2y-  2 

the  axes  are  rectangular,  the  hyperbola  is  equilateral,  as  also 
in  Ex.  2. 

4.  A  line  of  fixed  length  moves  with  its  extremities  in  two 
rectangular  axes.     Find  its  envelope. 

Let  AB  (Fig.  33)  be  the  line.     Its  equation  is 

-  +  ^  =  1,    or  u=bx  -{-  ay  —  ab  =  0,  (1) 

and  by  condition, 

a^  +  b^=AB'=P,  (2) 

I  being  constant.     Proceeding  as  before,  we  should  eliminate 
one  of  the  parameters  from  (1)  by  means  of  (2)  and  then  form 
the  partial  derivative.     But  it  will  be  found  more  expeditious 
to  differentiate  first  and  eliminate  afterwards! 
We  have  from  (1),  since  6  is  a  function  of  a, 

m    dU       '  db    ,  ,  ,  rt    .  .  T  /x  /ov 

since  ;y-  =  —  r  from  (2) .     Substituting  in  succession  the  values 
of  X  and  y  from  (1)  in  (3),  we  find 

a'y-\-bhj-b'  =  0,  (4) 

-  a'x  -  b^x  +  a?  =  0.  (5) 

Substituting  from  (2)  the  value  of  a^  in  (4)  and  of  b^  in  (5), 
6«  a" 

or         b^  =  yh\    a^  =  x¥, 

which  in  (2)  give  cc'  +  t/"^  =  l^,  the  equation  of  the  hypocycloid. 


136  THE   DIFFERENTIAL   CALCULUS. 

5.  Find  the  envelope  of  y  =  mx  +  b,  m  being  the  variable 
parameter. 

6.  From  a  point  A  on  the  axis  of  X  distant  a  from  the 
origin  lines  are  drawn.  Find  the  envelope  of  the  perpendic- 
ulars drawn  to  these  lines  at  their  intersections  with  Y. 

A  line  through  Ais  y  =  m{x  —  a),  and  its  intersection  with 
Y  is    (0,  —ma).      The   perpendicular    to    y  =  m(x  —  a)    at 

1  X 

(0,  —  ma)  is  y  •\-  ma  = x.     Hence  u  =  y  -{-  ma  -f  —  =  0,  in 

m  m 

which  m  is  the  variable  parameter. 

dm  m-       '  ^5 

Substituting  this  value  of  m  in  y  +  ma  +  —  =  0,  we  have 
y-  =  4  ax,  a  parabola. 

7.  Find  the  envelope  of  a  series  of  equal  circles  whose 
centres  lie  in  the  circumference  of  a  given  circle. 

Let  Xi^  -f  2/i^  =  ^1^  be  the  fixed  circle.     Then 

(x  -  Xi)- -\- {y  -  yi)- =  R^ 

is  the  movable  circle. 

Ans.  3?  -\-y'^  =  (i?i  ±  liy,  two  concentric  circles  Avhose  radii 
are  ^i  +  ^  and  R^  —  R. 

8.  Find  the  envelope  of  x cos 39  +  y&\\\'Sd  =  a (cos 2 ^) ^,  6 
being  the  variable  parameter. 

3 

xcos3^  +  t/sm3e  =  a(cos2e)^,  (1) 

whence  —  =  —  a;siii3e+  r/cos3tf  +  «(cos2e)^sin2e=  0, 

Old 

1 
or  a;sin3fl— ?/cos3e=  asin2e(cos2^)^.  (2) 

Squaring  (1)  and  (2)  and  adding, 

x2  +  J/2 ::::,  a2[(cos  2  d)^  +  COS  2  ^(siu  2  «)2]  =  oP'  cos2  fl.  (3) 


EVOLUTES   AND   ENVELOPES. 


137 


Dividing  (2)  by  (1), 


•    o  /.  o  /I        tan  3  fl  —  - 

%  sm  3d  —  y  cos  6  6  x 


xcos3e  +  y  sin  -id  y 

1  +  tan  Se-  - 

X 


=  tan  2  9, 


whence  ?'  =  tan  ^.     Hence  from  (3), 

X 

„  ,    2      o  1  —  tan2  e    „2^^  —  y^ 

x^  +  y^=  a^ —  =a  ^  — — ^, 

1  +  tan'^  fl  x2  +  j/2 

or  {x?  +  j^2)2  _  a2(j;2  _  j^2)^  the  lemniscate. 

108.    The  evolute  is  the  envelope  of  the  normals  to  the  involute. 
I 

Let  (if',  y')  be  any  point  P'  of  the  involute,  (a,  ft)  the  cor- 
responding point  Q  of  the  evolute,  and  (j>  the  angle  made 
by  the  normal  or  radius  of  curvature 
p  =  FQ  with  X.  Then  for  >SQ  and  ^ 
SF  we  have 

a— a;'=pcos<^,     p—y'=psm<f), 

or     a=x'+pCOS<f),     (3 = y'-]-p  sin  (f>.     (!)     o 

As  (x',  y')  moves  along  the  involute, 
(a,  fi)  moves  along  the  evolute,  or  a,  ^3,  y'  are  functions  of  x' 
Hence,  differentiating  (1), 

da  =  dx'-\-  cos  <t>dp  —  p  sin  <f>d(f>, 
dft  =  d7j'-\-  sin  <l>dp  +  p  cos  <f>d(f>. 

But,  Art.  26, 

dx'  =  sin  (f>ds,     dy'=  —  cos  <^fZi-, 

1      d<f> 
or,  since  -  =  -^-j 

da;'=  p  sin  <fid(}>,     dy'=  —  p  cos  (f)dcf). 
Substituting  these  in  (2),  we  have 

da  =  cos  ct>dp,     dp  =  sin  <f>dp,  (3) 

whence  -^  =  tan  ^. 
da 


(2) 


138  THE  DIFFERENTIAL  CALCULUS. 

But  —  is  the  slope  of  the  tangent  to  the  evolute  at  Q,  and 

da 

tan  <^  is  the  slope  of  the  normal  to  the  involute  at  P'.  Hence 
the  normal  to  the  involute  is  tangent  to  the  evohite,  and  the  evolute 
is  tlie  envelope  of  the  normals  to  the  invobite. 

109.  The  difference  between  any  two  radii  of  curvature  to  the 
involute  is  equal  to  the  arc  of  the  evolute  ivhich  they  intercept. 

For,  from  Art.  108,  Eq.  3, 

da  =  cos  (jidp,    dft  =  sin  cf>dp. 

Hence,  squaring  and  adding, 

da-+d/3'=dp'; 

or,  if  s'  be  the  arc  of  the  evolute  (Art.  25), 

ds'z=±dp; 

or  the  rates  of  change  of  s'  and  p  are  equal. 

110.  The  two  preceding  properties  afford  the  following 
mechanical  construction  of  the  involute  when  the  evolute  is 
given.  Let  ES  be  any  curve.  Then, 
if  a  pattern  of  RS  be  made,  and  a 
string,  one  end  of  which  is  fixed  at  S,  p,/_  ^'3-  35. 
be  wrapped  around  the  pattern  SQB, 
as  the  string  is  unwound  from  the 
pattern  the  free  end  will  describe  the  \i/ 
curve  MN  which  will  be  the  involute  of  MS.  Any  point  of 
the  string  will  trace  the  arc  of  an  involute  as  the  string  un- 
winds from  the  evolute ;  hence,  while  a  curve  has  but  one 
evolute,  namely,  the  locus  of  its  centre  of  curvature,  the  evo- 
lute has  an  infinite  number  of  involutes. 

111.  Orders  of  contact. 

Let  y=f{x),  y=(fi(x),  be  the  equations  of  two  curves  re- 
ferred to  the  same  axes  and  having  a  common  point  at  xz=  a. 


E VOLUTES  AMD  ENVELOPES.  139 

Then  /(a)  =  <f>{a).  Let  ^  be  a  very  small  increment  of  a,  the 
ordinates  corresponding  to  a;  =  a  +  ^  being /(a  +  7i),  <j>{a  -\-  h). 
By  Taylor's  formula, 

/(a  +  /0=  /(a)+/'(a)/i  +/"(«)  1+  /'"(a)  |  .••, 

<^(a  +  /0=<^(a)  +  <^'(a)/i  +  <^"(«),T;+<^"'(a),|'-, 
or,  by  subti-action, 

f{a+h)  -  «/,(a  +  /0  =  [/'(a)  -  <^'(«)]'^+  [/"(«)  -  <^"(«)]  ,| 

+  [/"'(a)-<^"'(a)]f^  +  -,(l) 

which  is  the  difference  between  corresponding  ordinates  of  the 
curves  on  one  or  the  other  side  of  their  common  ordinate 
according  as  h  is  positive  or  negative.  It  thus  appears  from 
(1)  that  two  curves  are  nearer  on  each  side  of  their  common 
point  as  the  second  member  is  smaller,  that  is,  as  the  succes- 
sive derivatives  in  order  are  equal  each  to  each  when  x  =  a. 

If  /'(a)  =  </>'(a),  the  curves  are  tangent  at  a;  =  a  and  are 
said  to  have  contact  of  the  first  order.    If,  also,  /"(«)  =  <^"(«)) 

the  curves  are  said  to  have  contact  of  the  second  order ;  and  so  on. 

■• 

Cob.  1.  Since,  if  the  curves  have  a  common  point,  we  must 
have /(a)  =  ^(a),  contact  of  the  nth  order  imposes  n  + 1  condi- 
tions. 

Cor.  2.  If  contact  is  of  an  odd  order,  the  first  term  of  (1) 
which  does  not  vanish  contains  an  even  power  of  h,  and  the 
difference  between  the  ordinates  has  the  same  sign  whether  h 
be  positive  or  negative.  Hence  one  curve  lies  above  or  below 
the  other  on  both  sides  of  the  common  ordinate,  or  curves  whose 
order  of  contact  is  odd  do  not  intersect.  If  contact  is  of  an  even 
order,  the  first  term  of  (1)  which  does  not  vanish  contains  an 
odd  power  of  h,  and  the  difference  between  the  ordinates 
changes  sign  with  h.  Hence  if  one  curve  lies  above  the  other 
on  one  side  of  the  common  ordinate,  it  lies  below  it  on  the 
other  side,  or  curves  whose  order  of  contact  is  even  intersect. 


140  THE   DIFFERENTIAL   CALCULUS. 

Cor.  3.  Since  the  number  of  independent  conditions  which 
can  be  imposed  upon  a  curve  is  the  same  as  the  number  of 
arbitrary  constants  in  its  equation,  the  highest  possible  order 
of  contact  between  two  curves  whose  general  equations  contain 
n  and  m  arbitrary  constants  is  w  —  1,  n  being  less  than  m. 

Examples.  1.  What  is  the  highest  possible  order  of  con- 
tact of  an  ellipse  and  parabola  ? 

The  general  equation  of  the  conies  contains  five  arbitrary  con- 
stants, and  therefore  the  ellipse  has  a  possible  fourth  order  of 
contact  with  other  curves.  But  for  the  parabola  e  =  1,  the 
number  of  arbitrary  constants  is  four,  and  its  highest  possible 
order  of  contact  is  the  third.  Hence  the  ellipse  and  parabola 
cannot  have  contact  with  each  other  above  the  third  order. 

2.  Prove  that  in  general  the  highest  possible  order  of  con- 
tact of  a  straight  line  is  the  first,  that  is,  tangency ;  and  of  the 
circle,  the  second. 

3.  Prove  that  at  a  point  of  inflexion  the  straight  line  has 
contact  of  the  second  order,  and  intersects  the  curve. 

At  a  point  of  inflexion  the  second  derivative  of  y  =f(x),  the 
equation  of  the  curve,  is  zero  (Art.  96).  Also,  from  y  =  mx  -{-  b, 
the  second  derivative  is  zero.  Hence  the  line  and  the  curve 
have  contact  of  the  second  order.  Hence,  also,  the  tangent 
intersects  the  curve  (Art.  Ill,  Cor.  2). 

4.  Prove  that  at  a  point  of  maximum  or  minimum  curvature 
the  circle  of  curvature  has  contact  of  the  third  order. 

At  such  a  point  the  circle  does  not  intersect  the  curve  (Art. 
103),  hence  its  contact  must  be  of  an  odd,  and  therefore  of  the 
third,  order  (Art.  Ill,  Cor.  2). 

SINGULAR    POINTS. 

112.  Points  of  a  curve  presenting  some  peculiarity,  inde- 
pendent of  the  position  of  the  axes,  are  called  singular  points. 
Such  are  points  of  inflexion,  already  considered  (Art.  103). 


SINGULAR   POINTS, 


141 


Fig.  36 


Fig.  37. 


<■ 


Multiple  points.  A  multiple  point  is  one  common  to  two  or 
more  branches  of  a  curve,  and  is  double,  triple,  etc.,  as  it  lies 
on  two,  three,  etc.,  branches. 

If  the  branches  pass  through  the  point, 
as  in  Figs.  36  and  37,  P  is  called  a  mul- 
tiple point  of  intersection  or  osculation, 
according  as  the  branches  have  different 
tangents  or  a  common  tangent.  Thus,  in 
Fig.  36,  P  is  a  triple  multiple  point  of 
intersection ;  and  in  Fig.  37,  P  is  a  double 
multiple  point  of  osculation. 

If  the  branches  meet  at  the  common 
point  but  do  not  pass  through  it,  as  in 
Figs.  38  and  39,  P  is  called  a  salient  point 
or  a  cusp  point,  according  as  the  branches 
have  difterent  tangents  or  a  common  tan- 
gent. Cusp  points  are  of  the  first  or 
second  species  according  as  the  branches 
lie  on  opposite  sides  or  on  the  same  side 
of  the  common  tangent. 


113.   A  conjugate,  or  isolated  point,  is 

one  whose  coordinates  satisfy  the  equa- 
tion of  the  curve,  although  no  branch  of 
the  curve  in  the  plane  of  the  axes  passes 
through  it ;  as  P,  Fig.  40. 

A  stop  point  is  one  at  which  a  single 
branch  of  a  curve  terminates. 


Fig.  40. 


114.   Determination  of  singular  points  by  inspection. 

Ascertain  if  possible,  by  inspection  of  the  equation,  whether 
for  any  value  of  one  of  the  variables,  as  x,  y  has  a  single  value. 
Let  a;  =  a  be  the  value  of  x  which  gives  a  single  value  h  for  y. 
Then  the  point  (a,  h)  is  to  be  examined. 

If,  for  values  of  x  both  a  little  less  and  a  little  greater  than 
a,  y  has  more  than  one  real  value,  the  branches  pass  through 


142 


THE   DIFFERENTIAL   CALCULUS. 


(a,  b),  which  is  therefore  a  multiple  point  of  intersection  or 
osculation.  If,  for  values  of  x  a  little  greater  (or  less)  than 
a,  y  has  more  than  one  real  value,  but  is  imaginary  when  x  is 
a  little  less  (or  greater)  than  a,  the  branches  meet  but  do  not 
pass  through  (a,  6),  which  is  therefore  a  salient  or  cusp  point. 
If  y  is  imaginary  for  values  of  x  both  a  little  less  and  a  little 
greater  than  a,  {a,  b)  is  a  conjugate  point. 

To  determine  whether  the  branches  have  the  same  tangent 
or  different  tangents  at  (a,  b),  we  observe  that,  since  (a,  b)  is 

common  to  several  branches,     '^  must  at  that  point  have  sev- 

dx 

eral  values,  and  the  branches  will  have  different  tangents  or 
a  common  tangent  according  as  these  values  are  different  or 
equal. 

It  is  evident  that  -^,  as  derived  from/(a^,  y)  =  0,  may  have 

more  thail  one  limit  when  f(^x,y)  =  0,  has  multiple  points. 
Thus  if  POP'  is  the  locus  of  f{x,  y)  =  0, 

-^,  being  entirely  general,  applies  to  both 

the  branch  OP  and  the  branch  OP',  and 

its  value  at  0  is  the  limit  of  ^  or  of  ^ 

X  x' 

according  as  P  or  P'  approaches  0 


a-'o 


Fig.  41. 

It  is  thus  a  general  ex- 
pression for  the  limits  of  different  ratios,  and  these  limits  may 
or  may  not  be  the  same. 


Examples.  1.  Prove  that  y-=  af{l  —  ar')  has  a  double  mul- 
tiple point  of  intersection  at  the  origin. 

Values  of  x,  whether  positive  or  negative,  give  in  general 
two  values  of  y ;  but  when  x  =  0,  y  has  the 


single  value  0.      Hence  the  branches  pass 
through  the  orgin. 

dy_  l-2a; 

dx 


=  ±1; 


Fig.  42. 


Vl  -  x'_ 

there  are  therefore  two  tangents  at  the  origin,  making  angles 
of  45°  and  135°  with  X,  and  the  branches  intersect. 


SINGULAR   POINTS. 


143 


2.  Prove  that  ah/—2ah^y  —  a'^=  0  has  a  point  of  osculation 
at  the  origin. 

Solving  /  for  y  we  have 


^ 


2/  =  ^(6±Vaa;  +  6-). 

If  X  is  positive  and  very  small,  the  radical  >6;  hence  one 
value  of  y  is  positive  and  the  other  nega- 
tive. If  X  is  negative  and  very  small,  both 
values  of  y  are  positive,  since  the  radical  is 
then  less  than  h.  li  x  =  0,  y  —  0.  Hence 
the  branches  pass  through  the  origin  and 
lie  in  the  second  angle  on  the  left  of  Y", 
and  in    the  first  and  fourth  on  the  right  of  Y. 

20  x"  +  4  aby  + A  ahx^^ 
dy      5x*-\-4abxy 

0,0 


Fig.  43. 


dx 


dx     2  a^y  —  2  abx^_ 


dy 


2a''^y 


dx 


—  4  abx 


0 


5^ 

dx 


whence  ^  =  ±  0,  or  the  axis  of  X  is  a  common  tangent  at  the 

dx 
origin.     Hence  there  is  a  double  point  of  osculation  at  the 
origin,  and  for  one  branch  the  origin  is 
a  point  of  inflexion. 

3.  Prove  that  y^=2x'^-\-x^  has  a  mul- 
tiple point  of  intersection  at  the  origin, 
the  tangent  having  the  slopes   ±  V2. 


4.  Prove  that  y^=    „^      has  a  double 
ar—x^ 

multiple    point   of   osculation   at    the 
origin. 

y  has  in  general  two  real  values  with 
opposite  signs,  whether  x  be  positive 
or  negative,  and  is  zero  when  a;  =  0 ; 
hence  the  branches  pass  through  the 
origin. 


Fig.  45. 


144 


THE   DIFFERENTIAL   CALCULUS. 


dy  _      2  g-  —  cr ' 


=  ±0. 


Hence  the  axis  of  X  is  a  common  tangent  at  the  origin, 

5.  Prove  that  y-  =  x'^  —  af  has  a  double  multiple  point  of 
osculation  at  the  origin.  The  locus  of  Ex.  4  is  represented  by 
the  dotted  line  of  Fig.  45,  and  that  of  Ex.  5  by  the  full  line. 

6.  Prove  that  the  cissoid  has  a  cusp  of  the  first  species  at 
the  origin. 

y- = If  X  is  positive,  y  has  two  values  with  oppo- 
site signs ;  if  cc  =  0,  y  =  0;  if  cc  is  negative,  y  is  imaginary. 
Hence  branches  in  the  first  and  fourth  angles  meet  at  the 
origin,  but  do  not  pass  through  it,  and  the  origin  is  either  a 
salient  point  or  a  cusp  of  the  first  species. 


dy  _    J    3  a  — X 

^^  (2a-x)'0> 


=  ±0, 


or  the  branches  have  the  axis  of  X  for  a  common  tangent. 

7.  Prove  that  ay^  =  x^  has  a  cusp  point  of  the  first  species 
at  the  origin. 

8.  Prove  that  (y  —  x^y  =  .r'  has  a  cusp 
point  of  the  second  species  at  the  origin. 

y  =  a^  ±  x'^.  If  X  is  negative,  y  is  imagi- 
nary ;  ii  X  =  0,  y  =  0 ;  if  a;  is  positive  and 
small,  y  has  two  positive  values.  He^ce 
two  branches,  both  in  the  first  angle  in 
the  vicinity  of  the  origin,  meet  at  the 
origin  but  do  not  pass  through  it. 

Hence  X  is  a  common  tangent,  and  the  origin  is  a  cusp  of  the 
second  species. 


V 

SINGULAR   POINTS.  145 

9.  Show  that  y^=x(a-\-xy  has  a  conjugate  point  at  (—a,  0). 
y  =  y/x(a  +  x)  has  two  values  when  x  is   positive,  but   is 

imaginary  for  all  negative  values  of  x  except  x=  —a,  when 
y  =  0. 

10.  Prove  that  the  conchoid  has  a  multiple  point  of  inter- 
section, a  cusp  of  the  first  species,  or  a  conjugate  point,  at 
(0,  —  b),  according  as  a  >  &,  a  =  &,  a  <  5. 

y-{-b     

x^y^  =  {y  •{■by{a? —  y^),  whence  x  —  ± va^  — y^- 

If  a  >  6,  values  of  2/  a  little  less  or  greater  than  —  h  give 
two  values  of  x,  and  cc  =  0  when  y=  —  h.  Hence  the  branches 
pass  through  (0,  —  6). 

If  a  =  b,  X  is  imaginary  if  y  is  negative  and  numerically 
greater  than  6 ;  is  0  when  y=  —  b;  and  has  two  values  when 
y  is  negative  and  numerically  less  than  b.  Hence  the  branches 
meet  at  (0,  —  b),  but  do  not  pass  through  it. 

If  a  <  &,  all  negative  values  of  y  numerically  greater  than  a, 
except  y  =  —  b,  render  x  imaginary. 


dy  fx 


dx      —a^y  +  a^y  —  27f  +  ba^  —  3by'' 


-&'2/J.,o 


2y.%  +  f 


-  2xy  +  {cr-x'-6y--6by-  b')  ^ 


0,0 


b- 


(a'-.)| 


^=± 


dx  Va'  -  &- 

If  a>  b,  there  are  two  tangents  who  slopes  are  ± 


/«.2  . 


Va 


If  a  =  b,  the  slope  becomes  00,  and  F  is  a  common  tangent. 
it  a  <  0,  -p  IS  imaginary. 


146 


THE   DIFFEKENTIAL   CALCULUS. 


11.  Show  that  y  =  ictan~^-  has  a  salient  point  at  the  origin. 


If  a;  =  0,  y  =  0;  whether  x  be  positive 
or  negative,  y  is  positive.  The  curve 
therefore  lies  above  X,  and  branches  in 
the  first  and  second  angles  meet  at  the 
origin.     When  x  is  positive, 


dy 


=  tau~'- 
dx  X 


= ""  =  1.5708, 


l+a^_ 
the  slope  of  the  branch  in  the  first  angle.    When  x  is  negative, 


^^  =  tan-' 
dx 


+ 


=  tan  (-  X  )  =  -  ^  =  -  1.5708, 


1  +  X- 
the  slope  of  the  branch  in  the  second  angle. 

12.  Prove  that  y  =  x  log  x  has  a  stop  point  at  the  origin. 

The  curve  lies  to  the  right  of  Y,  for      y 
negative   numbers  have  no  logarithms, 
and  X  cannot  be  negative.     When  x  is 
positive,  y  has  one  real  value.     When      o- 
x  =  0, 


y  =  x  log  X  = 


11 

logic 

1 

X 

X 
0                •*'"_ 

Fig.  48. 


-;r.  =0. 


Hence  the  curve  consists  of  a  single  branch  terminating  at 
the  origin. 


ASYMPTOTES. 

115.  A  rectilinear  asymptote  to  a  curve  is  a  straight  line 
which  the  curve  continually  approaches  but  never  reaches ;  or 
it  may  be  defined  as  the  limiting  position  of  the  tangent  as  the 
point  of  contact  recedes  indefinitely  from  the  origin. 

If  the  curve  has  no  infinite  branch,  it  can  have  no  asymptote. 


asy:mptotes.  147 

116.   Asymptotes  parallel  to  the  axes. 

If  PQ  is  an  asymptote  parallel  to  X,  and  at  a  distance  b  from 
it,  then  as  x  increases  without  limit,  y  approaches  the  finite 
limit  b,  and  y  =  b  is  the  equation  of  PQ. 
So,  also,  if  SR  is  an  asymptote  parallel  to 
Y,  and  at  a  distance  a  from  it,  then  as  y 

increases  without  limit,  x  approaches  the    p_  < i .- 

finite  limit  a,  and  x  =  a  is  the  equation 
otSR. 

To  determine,  therefore,  whether  f{x,  y)  =  0  has  asymptotes 
parallel  to  the  axes,  observe  whether  either  variable  approaches 
a  finite  value  as  a  limit,  that  is,  as  the  other  increases  indefi- 
nitely. If  such  be  the  case,  there  is  an  asymptote  parallel  to 
the  axis  corresponding  to  the  variable  which  increases  indefi- 
nitely, at  a  distance  from  it  equal  to  the  corresponding  finite 
limit  of  the  other  variable. 

Examples.  1.  Show  that  x=2R  is  an  asymptote  to  the 
cissoid. 

y-  —  —— ,  in  which  y  approaches  ±  oo  as  a;  approaches 

2  R  —  X 

2R.     Hence  x  =  2R  is  an  asymptote  to  both  branches. 

2.  Show  that  y  =  0  is  an  asymptote  to  the  conchoid. 

x  =  ±  --i-Z—  Va-  —  y\  in  which,  whether  y  be  positive  or  neg- 
ative, X  approaches  ±  co  as  y  approaches  0.  Hence  y  =  0,  or 
the  axis  of  X,  is  an  asymptote  to  both  the  branch  above  and 
that  below  X. 

3.  Examine  y  =  tan  x  for  asymptotes. 

4.  Show  that  r/  =  0  is  an  asymptote  to  the  witch 

x^y  =  4:R'-{2R-y). 


148 


THE   DIFFERENTIAL   CALCULUS. 


5.  a-y  —  x-y  =  a^. 


y  =  — -•     As  X  approaches  ±cc,  y 

approaches  0.  Hence  y  =  0,  or  the  axis 
of  X,  is  an  asymptote  to  two  branches. 
Also,  y  approaches  oo  as  x  approaches 
±  a.  Hence  x  =  a  and  x  =  —  a  are 
asymptotes. 


6.  a-x  =  y{x  —  aY 


y 


a'x 


As  x  approaches  ±  oo, 


{x-ay 
y  approaches  ±  0.      Also  y  approaches 
CO  as  cc  approaches  a.     Hence  the  axis  of  X  and  x  =  a  are 
asymptotes. 

7.  xy  —  ay  —  bx  =  0. 

y  = ,  X  =  — - —     The  asymptotes  are  x  =  a,  y  =  b. 

X  —  a  y  —  b 

8.  Show  that  y  =  0  is  an  asymptote  to  x  =  log  y. 

9.  Examine  aPy^  =  a-{x^  —  y^)  for  asymptotes.    Ans.  y—  ±a. 

10.  Examine  y{a-  -\-  x-)  =  a^{a  —  a;)  for  asymptotes. 

Ans.  w  =  0. 

b^ 

11.  Examine  y  =  a-\ for  asymptotes. 

{x  —  cy 

Ans.  y  =  a,  x  =  c. 

12.  Examine  the  locus  of  Ex.  4,  Art.  114,  for  asymptotes. 

117.   Asymptotes  oblique  to  the  axes. 
The  equation  of  a  tangent  to  a  plane  curve  being 

dy' 

y-y'=dx'^^~^'^' 


if  we  make  in  this  equation  y  =  0,  the  corresponding  value  of 


ASYMPTOTES.  149 

X  will  be  the  intercept  of  the  tangent  on  X  Kepresenting 
this  intercept  by  X,  we  have 

X=x'-y'^-  (1) 

dy' 

In  like  manner,  making  x  =  0,  the  intercept  on  Y  is  found 
to  be 

Y=y'--'%'  (2) 

from  which  the  accents  may  be  omitted  if  we  understand 
(x,  y)  is  the  point  of  tangency.  Now  the  asymptote  is  the 
limiting  position  of  the  tangent,  that  is,  the  position  which 
the  tangent  approaches  as  the  point  of  contact  recedes  indefi- 

dy 
nitely ;  hence  its  slope  is  the  limit  of  -^,  and  its  intercepts  are 

the  limits  of  (1)  and  (2),  as  the  point  of  contact  recedes  in- 
definitely from  the  origin.  The  position  of  the  asymptote 
when  oblique  to  the  axes  will  therefore  be  known  when  the 
limits  of  X  and  Y  are  known,  and  if  these  limits  be  designated 
by  Xi  and  Yj,  the  equation  of  the  asymptote  is 

Xj  Yr 

If  either  Xi  or  Yi  is  zero,  the  asymptote  passes  through  the 
origin,  and  its  direction  is  determined  by  finding  the  limit  of 

dv 

—  as  the  point  of  contact  recedes  indefinitely  from  the  origin. 

If  both  Xi  and  Ti  are  infinity,  there  is  no  asymptote.  If  one 
is  infinity  and  the  other  finite  or  zero,  the  asymptote  is  parallel 
to  or  coincides  with  the  axis  on  which  the  intercept  is  infinite. 

dy 
It  is  usually  most  expeditious  to  find  first  the  limit  of  ^• 

If  this  is  neither  0  nor  co,  the  asymptote  is  oblique,  and  its 
position  is  made  known  by  either  Xi  or  Yi;  if  the  limit  of 

dv 

~  is  zero,  there  will  be  an  asymptote  parallel  to  the  axis  of 

X  if  Fi  is  finite ;  if  the  limit  of  ^  is  oo,  there  will  be  an 
asymptote  parallel  to  the  axis  of  Y  if  Xi  is  finite. 


150 


THE   DIFFERENTIAL   CALCULUS. 


Examples.     Examiue  the  following  curves  for  asymptotes. 

1,  The  parabola,  y^  =  22)x. 

The  curve  has  infinite  branches  in  the  first  and  fourth  angles. 

-^  =  -       =  ±  0 ;  hence  if  there  are  asymptotes,  they  are  paral- 

lei  to  the  axis  of  X.      Yi  =  y  —  x 
there  are  no  asymptotes. 


'_)»,    ±0 


=  ±  X ;  hence 


2.  The  hyperbola,  ary^  —  6V  =  —  a-6l 

The  curve  has  an  infinite  branch  in  each  angle. 


dx 


X.= 


b'x       ,   b    L    ,  b- 
ay  a  ^        y^ 

dx~\  ( 


=  ± 


=  ±0. 


Hence  the  diagonals  of  the  rectangle  on  the  axes  are  asymp- 
totes to  the  curve  in  each  angle. 

3.  x  =  log  y. 

Since  y  approaches  0  as  a;  approaches  —  x,  the  axis  of  X  is 


an  asymptote  (Art.  142).     Otherwise, 


dy 
dx 


y 


=  0  ;  hence  if 


there  is  an  asymptote  to  the  branch  in  the  second  angle,  it  is 
parallel  to  X. 

1 
1  -  log  y 


yi  =  2/-2/logy]o 


=  0, 


or  the  axis  of  X  is  the  asymptote. 

For  the  branch  in  the  first  angle,  a;  =  oo  when  y  =  x.    Hence 

dy 

-f-  =  y     =  00  ;  that  is,  the  asymptote  is  perpendicular  to  the 
dx       J^ 

axis  of  X,  if  one  exists.      Xi  =  a;  —  1]^  =  oo,  or  there  is  no 

asymptote  to  the  branch  in  the  first  angle. 


•CtfRVE   TRACING. 


151 


4.    y^  =  x^(a  —  x). 

When  X  >  a,  y  is  negative,  and  there  is  an  infinite  branch  in 
the  fourth  angle.  When  x  is  negative,  y  is  positive,  and  there 
is  also  an  infinite  branch  in  the  second  angle. 


X,= 


2a-3a; 


Y,= 


Hence  the  asymptote  is  common  to  both  branches,  and  its 

equation  is  y  =  —  x-{--'     (See  Fig.  53.) 
o 

5.  Prove  that  y  =  x  +  2  is  an  asymptote  to  y'  =  6  a;^  -j-  x^. 

6.  Prove  that  r/=  —  ic  is  an  asymptote  to  y  =  a^  —  ar"'. 


CURVE    TRACING. 

118.  The  foregoing  principles  are  sufficient  for  the  deter- 
mination of  the  forms  and  singularities  of  many  curves,  but  a 
knowledge  of  the  general  theory  of  curves  is  necessary  in  order 
to  trace  curves  with  facility  from  their  equations. 

l-x" 


1.  y  = 


1  +  ar^ 


2a;(a^-3) 


(3). 


(2), 


{l+x'Y 

Since  y  has  but  one  value  for  any  value  of  x,  its  sign  being 
that  of  x,  and  is  0  when  y  =  0,  the  curve  passes  from  the  third 
to  the  first  angle  through  the  origin, 
and  has  infinite  branches  in  these 
angles.  As  x  approaches  ±  cc,  y 
approaches  0,  and  the  axis  of  X  i's 
therefore  an  asymptote  to  both 
branches.  f'{x)  changes  sign  at 
x  =  ±l,  and  these  values  render  f"{x)  negative  and  positive 
respectively,  giving  a  maximum  ordinate  in  the  first,  and  a 
minimum  ordinate  in  the  third,  angle.    f"{x)  changes  sign  at 


Fig.  52. 


152 


THE   DIFFERENTIAL   CALCULUS. 


x=±y/S  and  a;  =  0,  giving  three  points  of  inflexion.  The 
slope  of  the  curve  at  the  origin  is  45°,  for  /'(x)  =  l  when 
x  =  0. 


2.  y'  =  aa^  —  of'     (1), 


/"(■v)  = 


/'(•«)  = 


2a -3a; 


2  a" 


9.x^(a-a;)^ 


3x^{a 
(3). 


xy 


(2), 


If  X  is  negative,  y  is  positive,  and  there  is  an  infinite  branch 
in  the  second  angle.  f"{x)  is  negative  when  x  is  negative, 
hence  this  branch  is  convex  upward. 

If  X  is  positive,  y  is  positive  till 
x  =  a,  when  y  =  0,  the  curve  having  a 
branch  which  crosses  X  at  x  =  a  from 
the  first  into  the  fourth  angle.  Since 
y  =  0  when  a;  =  0,  the  branches  meet 
at  the  origin,  Avhich  is  a  cusp  point  of 
the  first  species, /'(a;)  becoming  co  for 
x  =  0,  and  Y  being  the  common  tan- 
gent. f"{x)  changes  sign  from  —  to  +  at  a;  =  a,  which  is 
therefore  a  point  of  inflexion,  the  curve  being  convex  upward 
in  the  first  angle  and  concave  in  the  fourth.  The  slope  at 
x  =  a  is  00,  since  f  (x)  =  cc  when  x=a.  f'(x)  changes  sign 
at  x  =  ^a  from  +  to  — ,  hence  x  =  ^a  renders  y  a  maximum. 

It  has  been  shown  in  Art.  145,  Ex.  4,  that  y  =  —  a?  -f  -  is  an 

o 
asymptote  to  both  branches. 


3.  2/  =  e"«(l),    f'{x)  =  ±-^  (2),    f"(x)  =  -^^  (3). 

x'e'  x'^e" 

From  (1)  we  observe  that  y  is  positive  whatever  the  value 
of  X,  or  the  curve  lies  above  X. 

Let  X  be  negative.     Then  y  =  e^,  which  increases  as  x  de- 
creases, becoming  ac  when  a;  =  0 ;  and  decreases  as  x  increases, 


CURVE  TRACING. 


153 


Fig.  54. y 

r 

^,/"'^ 

0 

X 

becoming  1  when  x  =  v:.  Hence  y  =  1,  and 
the  axis  of  Y,  are  asymptotes  in  the  second 
angle.  Also,  when  x  is  negative,  f"{x)  is 
positive,  and  this  branch  is  concave  upward. 

Let  X  be  positive.      Then  y=  — ,  which 

increases  with  x,  becoming  1  when  a;  =  x  ;  or,  y  =  1  is  also  an 
asymptote  to  the  branch  in  the  first  angle.  Since  y  =  0  when 
a;  =  0,  and  y  =  '-c  when  x  is  negative  and  very  small,  the  origin 
is  a  stop  point. 

f'{x)  cannot  change  sign,  lience  there  are  neither  maxima  nor 
minima  ordinates. 

f"{x)  changes  at  a;=  ^  from  -f-  to  — ,  a  point  of  inflexion  at 
which  the  cnrvature  changes  from  concave  to  convex  upward. 


f'i^)  =  —. 


=  0  X  cc.     Placing  z  =  -,  whence  z  =  x  when 


are'-'  ,_ 
^  =  0, /'(a-)  =  -; 
orisrin. 


e' 


=  0,  and  X  is  a  tangent  at  the 


4.  y  =  x\ogx  (1),    /'(a-)  =  l  +  logx   (2),    /"(^)  =  ^  (3). 

The  curve  lies  to  the  right  of  Y,  since  x  cannot  be  negative. 
As  the  logarithm  of  a  proper  fraction  is  negative,  y  is  negative 
till  a;  =  1 ,  when  y  =  0.  When  x>l,  y  is 
positive.  As  f"{x)  cannot  change  sign,  ^'' 
the  curve  is  concave  upward.     f'{x)  =  0 

gives  log  x  =  —  l,  or  x  =  ('~^  =  ',  which  ^\ 

renders    y    a    minimum.      When   x  =  0, 
f'(^x)  =  —  cc,  or  the  axis  of  1'  is  a  tan- 
gent.    When  x  =  l,  f'{x)=:l,  or  the  curve  crosses  X  at  an 
angle  of  4.5°. 


Fig.  55. 


^^  dx  X 

fly      1  4-  log  X 


=  T         =0C, 


Y,  =  y-x^  =  -x 
ax 


154 


THE   DIFFERENTIAL   CALCULUS. 


hence  there  are  no  asymptotes.     The  origin  is  a  stop  point 
(Art.  114,  Ex.  11). 


o.   {y  -  xf  =  aA  or  2/  =  r^  ±  x^  (1),  f\x)  =  x{2  ±  |V^)  (2), 

f"{x)  =  2±-'^-Vx  (3). 
See  Ex.  7,  Art.  114,  and  Fig.  40. 


G.  r  =  «^;c(l),       /'(,•)  =i^,  (2), 

oy- 

7.  (»r  =  r'(l),       /'(x-)  =  |^(2), 

2  ay 


./■"(x-)  =  f^,  (3). 
4  a^ir 


8.  ?/-"  =  2ar^  +  ^-'  (1)- 


4  +  3a; 


.r(x)=±     ^+3a-      ^3^^ 
4(2 +  x)^ 


2  V2  +  X 


(2), 


Erom  (1),  ?/=  ±  a;V2  +  a;,  from  which  we  see  that  the 
curve  is  symmetrical  to  X,  passes  through 
the  origin,  and  has  x  =  ~2,  x  =  oo  for  its 
limits  along  X.  f'(x)  =  ±  V2  when  x=0, 
hence  the  origin  is  a  multiple  point  of  inter- 
section. The  tangent  at  x=—2  is  perpen- 
dicular to  X,  since  f'{x)  =  oo  for  x  =  —  2. 
f"{x)  has  two  signs,  but  does  not  change 
sign  except  for  x  =  —  |,  which  is  not  a  point  of  the  curve,  since 
the  limit  of  re  is  —  2 ;  hence  there  are  no  points  of  inflexion. 

f'(x)  changes  sign  at  a;  =  — |,  where  there  is  a  maximum 
and  a  minimum  ordinate. 


Fig.  56. 


fi(   X  _  f^.y_4a5  + 3ar" 


4  -f  6  a;  ,            dy 

z — ,  whence  -^ 

()dy  ax 

dx 


or  the  asymptote,  if  there   be   one,  is   perpendicular   to    X. 


Xi  = 


4-f3a; 


cc,  and  there  is  no  asymptote. 


CURVE  TRACING. 


155 


9.  y'  =  x*  +  x^  (1),      /'(x)  =  ^(4-h5a.)=  ±  i^jtl^  (2), 
^,^^^^^8  +  24^+15^  (3).    ■ 


From  (1) ,  y=  ±  a^  Vl  +  x,  whence  we  see  the  curve  is 
symmetrical  to  X,  passes  through  the  origin, 
and  that  its  limits  along  X  are  x  =  —  l  and 
X  =  X.  When  x  =  0,  /'  (a?)  =  ±  0  ;  hence  the 
origin  is  a  point  of  osculation,  X  being  a 
tangent  to  both  branches. 

From  f"{x)  =  0,  we  find  x=  ""^^"^^^^ 


15 


;   the  lower  sign 


is  impossible  since  a;  =  —  1  is  a  limit,  and  the  upper  sign  gives 
points  of  inflexion.  f'{x)  =  0,  gives  x  =  0  and  a;=  —  4,  which 
correspond  to  maxima  and  minima  ordinates.  There  are  no 
asymptotes. 


10.  y-  =  x^  —  X*,  ov  y=±  xl^l  —  x  (1) 

3-4a; 


\x)  = =  ± 


2y 


/"(^)=± 


;a^_12a;4-3 


-X) 

(3). 


(2), 


Fig.  58. 


4(l-a;)Va;(l-a;) 
From  (1)  the  curve  is  seen  to  be  symmetrical  with  respect 
to  X;  and  as  x  cannot  be  negative  and 
/'(»)  =  0  when  x  =  0,  the  origin  is  a  cusp 
of  the  first  species.  Since  x  cannot  be 
greater  than  1,  the  curve  lies  between  the 
limits  0  and  1  along  X.  There  is  a  maxi- 
mum and  a  minimum  ordinate  at   a^  =  f , 

and   a;  =  '  ~"^'    corresponds  to  points    of   inflexion.     When 

4 
a;  =  l, /'(x)  =  x. 

11.  a-y-x'y^aJK  13.  a/ -  a^  +  &ar' =  0. 

12.  4a;  =  y{x  -  2)^  "  14.  x"'  -  ay  +  1  =  0. 


156 


THE    DIFFERENTIAL   CALCULUS. 

15.  f  =  a^(l  _  :^Y  (Fig.  59).        18.  y-  =  x*  -  .^■«  (Fig.  45). 

16.  f  =  x\l  -  :^Y  (Fig.  60) .       19.  f  +  x'  _  «« =  0. 

17.  dY  +  b'^x^  =  d^b^. 


Fig.  59. 


Fig.  60. 


POLAR   CURVES. 

119.    Subtangent  and  subnormal. 

Let  P  be  any  point  of  MM ',  PT  the  tangent,  PN  the  normal, 
0  the  pole,  and  OX  the  polar  axis.  Through  the  pole  draw 
NT  perpendicular  to  the  radius  vector 
to  the  point  of  contact,  OP,  meeting 
the  tangent  and  the  normal  at  T  and 
N.  Then  OT  is  the  subtangent  and 
OiVthe  subnormal. 


120.   Lengths  of  the  subtangent  and 
tangent. 

From  the  right  triangle  OPT, 

0T=  OP  tan  OPT=  rtan  OPT. 

tan  a  —  tan  0 


But  tan  OPT  =  tan  (a  -  ^)  = 


-r  —  tan 
ax 


1  +  tan  a  tan  6 
dy  cos  6  —  dx  sin  6 


1-f-^tan^      (^os  6 +  dy  sin  6 
dx 

But  x  =  rcos6,  y  =  rsin$,  whence 

dx  =  cos  ^dr  -  r  sin  ^d^,    (7y  =  sin  Odr  +  r  cos  Odd. 
Making  these  substitutions,  we  find  tan  OPT=r—-  hence 


Subtangent  =  OT=i" 


ode 
dr 


POLAR   CURVES.  157 


and  Tangent  =  V  OP'  -\-OT-=r^  \l  +  r"  ~ 


Cor.         ds  =  Vdx^  -\-  dy^  (Art.  25)  =  Vrfr'  +  iHG^. 

121.   Lengths  of  the  subnormal  and  normal. 

0N=  OP  tan  OPN=  root  OPT= 

tan  OPT 
Hence  (Art.  120), 

Subnormal  =  0A^=^^-', 
dO 


and  Normal  =  PN  =  ^  /  ?•'  +  -^'• 


Examples.     1.  The  lemnisoate  r^  =  a^  cos  2  B. 

dr _  _ci- sin 2 6 
dd~  r 

Hence      Subt  =  r-  --  = ~ =  -  r  cot  2  ^, 

dr  a'  sin  2  ^ 


Tangent  =  VylX  +  r^'^  =  --^^, 
\  di^      Va*  -  r* 

Subn  =  ^"=-^^!-?i2l^, 


Normal  =  ^  r'  +  ^^  =  —  • 

\  (7^         T 

2.  The  logarithmic  spiral /•=€<*. 
—  =  a*  losj:a. 

Hence      Subt  = ,     Subn  =  r  log  a. 

log  a 

dO         1 

In  this  spiral   tn.\\OPT=r —  = ,    a  constant.     Hence 

dr      log  a 

the  tangent  makes  a  constant  angle  with  the  radius  vector  to 

the  point  of  contact.    For  this  reason  this  spiral  is  often  called 


158 


THE  DIFFERENTIAL  CALCULUS. 


the  equiangular  spiral.     In  the  Naperian  logarithmic  spii-al, 
log  e  =  l,  and  the  subnormal  is  equal  to  the  radius  vector. 


3.  The  spiral  of  Archimedes  r  =  a$. 

—  =  a  —  subn,  a  constant.     1^ —  =  a€F  ■■ 
dO  dr 

4.  The  cardioide  r  =  (i{'l  -f  cos^). 

dr 


subt. 


dd 


z=  —  a  sin  6  =  subn.     Subt.  =  — 


a  sin  ^ 


5.  Prove  that  in  the  curve  r=a  sin  6  the  radius  vector  makes 
equal  angles  with  the  tangent  and  polar  axis. 


tan  OPT  =  r—  =  a  sin  0 — — 
dr  a  cos  6 


=  tan  6. 


6.  The  circle  r  =  2R  cos  0. 

Subt  =  2 Root  e cos  6,    subn  =  -2EsmO, 
Tangent  =  2R cos  6 cosec  6,   normal  =2R. 

7.  Prove  that  the  subtangent  of  the  reciprocal  spiral  is  con- 
stant. 


122.    Curvature  of  polar  curves, 

A  curve  at  any  of  its  points  is  said  to 
be  convex  or  concave  towards  the  pole 
according  as  its  tangent  does  or  does  not 
lie  on  the  same  side  of  the  curve  as  the 
pole. 

Let  fall  from  the  pole  the  perpendicular 
OD=p  upon  the  tangent.  If  the  curve 
is  concave  to  the  pole,  2^  is  an  increasing 
function  of  r,  r  =f{0)  being  the  equation 

of  the  curve.     Therefore  -^  is   positive. 

dr        ^ 

If  the  curve  is  convex  to  the  pole,  p  is 

a  decreasing  function  of  r,  and  ~^  is  nega- 

, .  dr 

tive. 


Fig.  62.  /p 


POLAR  CURVES.  159 

Hence  the  curve  is  concave  or  convex  towards  the  pole  ac- 
cording as  —  is  positive  or  negative,  and  at  a  point  of  inflex- 
dr 


ion  -t-  must  change  sign 
dr 


dp 

dr 
To  find  p,  we  have  (Fig.  63),  NP  being  the  normal, 

OD:NP::OT:  NT. 

But,  Arts.  120  and  121, 


NP=Jl^+—;  0T=')^'-^, 

\         de^  dr- 

NT=NO+OT=^  - '-  +  r' ^^^. 
dO         dr 


Hence  p  = 


V^+*^ 


de" 

To  examine  a  polar  curve  for  points  of  inflexion,  substitute 

—  from  the  equation  of  the  curve,  r  =f(0),  in  the  above  value 

of  p,  and  see  if,  for  any  value  of  r,  -^  changes  sign. 

dr 

Examples.     1.  Prove  that  the  logarithmic  spiral  is  always 
concave  to  the  pole. 

„  dr        ,  '/•-  r 

r=a'',     .-.  -3^=rloga,    p  = 


dO  lr^4.^      Vl+log^a 

\        d&' 

Hence  ^  — 


<ir      Vl  +  log^a 
which  is  always  positive. 

2.  Examine  the  lituus  for  curvature. 

_  _a_       .    dr  _  _    a  _       2  aV 


Hence  ^  =  2a^(i^.^)  =  0, 


160 


THE   DIFFERENTIAL   CALCULUS. 


gives    r  =  aV2.      If   ?'<rtV2,    ^   is   positive;   if   r>aV2, 

—  is  negative.     Hence  ?-=aV2  indicates  a  point  of  inflexion 
dr 

at   which  the   curvature   changes    from    concave    to    convex 

towards  the  pole. 

3.  Prove  that  the  parabola  r  =  — ^ is  always  concave 

to  the  pole. 


123.  Radius  of  curvature. 
From  Art.  119,  we  have 


P  = 


)^^h 


d^ 
dx" 


(1) 


in  which  a;  is  equicrescent,  and  the  problem  is  to  transform 
(1)  into  its  equivalent  in  terms  of  r  and  6  when  6  is  equi- 
crescent. 

Therefore  (Art.  58,  Ex.  7), 


P  = 


+  -r 


\d6J        dS" 


(2) 


Examples.     Find  the  radius  of  curvature  of : 
1.  The  lemniscate,  ?"  =  a^cos26. 


dr         a^  sin  2  ^ 
d6               r 

Va*  -  r' 
r 

dV          r*  -f  a* 

d^              f^ 

Hence 


P  =  , 


'dr 


POLAK   CURVES. 

2.  The  cardioide,  r  =  a{l—GosO). 

(Id 

—  =  acoiiO  =  a  —  r. 

Hence  p=  2V2  ar. 

3.  The  spiral  of  Archimedes,  r  =  aO. 

^       2a- +  ^^  2  +  0' 


161 


4.  The  reciprocal  spiral,  r  =  -• 


or  u* 

5.  The  lituus,  r  =  — =• 

Ve 

^~2a-     Aa*-r* 

6.  The  logarithmic  spiral,  r  =  a*. 

p  =  a«(l+log2a)^. 

If   a  =  e,   p  =  V2e*=rV2,   or  the  radius  of   curvature  is 
V2  X  the  radius  vector. 

124.  Asymptotes. 

Since  the  asymptote  is  the  limiting  position  of  the  tangent 
as  the  point  of  contact  recedes  indefinitely 
from  the  pole,  if  a  polar  curve  has  an  asymp- 
tote, r  must  be  infinite  for  some  finite  value 
of  6,  and  for  such  value  of  6  the  subtangent 

d6 
r^—  must  be  finite. 

dr 

Let  a  be  the  value  of  B  which  renders  r  in- 
finity.     To   construct   the   asymptote    make    ^jr 
AOP=a,  draw  through  0  a  perpendicular  to 


162  THE   DIFFERENTIAL   CALCULUS. 


OP,  and  make  OT=t^~ 
ilr 


.     Then  TQ,  parallel  to  OP,  is  the 

asymptote.      For   the   point   of  contact    being    infinitely    dis- 
tant from  0,  the  radius  vector  and  asymptote  are  parallel.     If 

=  cc,  there  is  no  asymptote. 


dr 


Examples.     1.  Examine  the  hyperbola  for  asymptotes. 

p  ,  dr  ewsin^  ■,    ul6         p 

r  = — ,  whence  —  = ^- —  and  r —  =  — ± 

ecos^  — 1  dQ      {eQ,o^B  —  \y  dr      esin^ 

Now  r  =  cc  when  cos  0  =  -==  —    '  Hence,  if  there  be 

an  asymptote,  it  is  parallel  to  the  diagonal 
of  the  rectangle  on  the  axes.     Again, 


e=Vl-  cos^  e  = 


Ve^'  -  1 


.dO 


hence  r^ —  =  — 4- —  =  ctVe^  —1  =  6. 
dr      e sin  6 

There  is  therefore  an  asymptote.  To  construct  it,  draw  OP 
parallel  to  the  diagonal  on  the  axes  (or  make  AOP=  cos"^- ), 
and  make  OT=h.  Then  TQ,  parallel  to  OP,  is  the  asymp- 
tote.    Since  OC  = ^  =  —  =  ae,  C  is  the  centre,  and 

sma      Ve-  — 1 
the  asymptote  coincides  with  the  diagonal.     Also,  as 

cos  9  =  cos  {—  0), 

there  is  another  asymptote  below  the  axis,  and  similarly  situ- 
ated. 

2.  Prove  that  the  parabola  r  = =- has  no  asymptote. 

3.  Prove  that  the  lituus  r  =  — -  has  the  polar  axis  for  an 
asymptote.  ^^ 


POLAR   CURVES.  168 


4.  Prove  that  the  spiral  r  =  -  has  an  asymptote  parallel  to 

6 
the  polar  axis  at  a  distance  a  from  it. 

^    1,.         ■         ..      (r'sinS^  ,.  ,    , 

o.  r^xamme  r  = tor  asymptotes. 

cos  6 

().  Examine   (r  —  a)  siu d  =h  for  asymptotes. 

7.  Examine  the  conchoid  r  =  h  cosec  6  +  a  for  asymptotes. 

r  =  c»  when  ^  =  0.     ?- —  = -^ — ! '--    =b. 

dr  b  cos  d       Jo 

Hence  the  asymptote  is  parallel  to  the  polar  axis,  and  at  a 

distance  from  it  equal  to  b. 

125.    Tracing  of  polar  curves. 

Write  the  ecpiation  of  the  curve  f(r,  $)  =  0  in  the  form 
r=f(6),  when  possible,  and  assign  such  values  to  9  as  will 
render  it  easy  to  determine  those  of  r.  This  will  usually  be 
sufficient  to  determine  the  general   form   of   the  locus.     For 

maxima  or  minima  values  t)f  r,  —  must   change  sign.     The 

do 

locus  may  then  be  examined  for  curvature,  points  of  inflexion, 
and  asymptotes. 

ExAMPLKs.     1.  '/•=asin2^.     —  =  2«cos2^. 

dd 

r  =  0  when  ^  =  0  and  increases  with  6  till  6  =  -,  when  — 

4  dd 

changes   sign  from  +  to  — .      Hence   6  =  -  renders   r  =  a  a 
maximum. 

From  6  ="  -  to  6  =  -  r  decreases  from  a  to  0,  and  the  curve 

is  a  loop  in  the  first  angle. 

When  $  passes  -,  r  becomes  negative,  in- 

2  (If, 

creasing  numerically  till  0  =  ^Tr,  when  — 

dd 

changes  sign  from  —  to  +,     Hence  ^  =  ^tt 


164 


THE   DIFFERENTIAL   CALCULUS. 


renders  r  =  —  a  a  minimum.  From  ^  =  f  tt  to  6  =  ir,  r  is  still 
negative,  but  decreases  numerically  from  a  to  0,  giving  another 
loop  in  the  fourth  angle. 

When  0  passes  ir,  r  becomes  positive,  and  as  sin  2  6  passes 
through  all  its  values  while  6  varies  from  0  to  tt,  equal  loops 
will  be  traced  for  values  of  6  between  ir  and  2  tt. 

The  maximum  and  minimum  values  of  r  are  derived  from 

-^=0;  namely,  Itt,  Itt,  frr,  and  In. 
cW 

Xo  value  of  6  renders  r  =  x  ;  hence  the  curve  has  no  asymp- 
tote. 


2.  r  =  a  sin  3  6. 

The  curve  is  shown  in  the  figure. 
From  this  example  and  Ex.  1  it  may 
be  inferred  that  in  all  equations  of 
the  form 

r  =  a  sin  ?i^, 

the  curve  consists  of  n,  or  of  2w,  loops, 
according  as  n  is  an  odd  or  an  even 
integer. 

3.  r  =  asin|(Fig.  68). 

4.  r=a(l-tan^)  (Fig.  69). 

5.  r  =  a  cos  2  6. 

6.  r  =  a  +  sin  \  6. 

7.  r=a  +  .sinf  ^  (Fig.  70). 

8.  r  =  a  +  tan  2  6. 

9.  y2  =  a-(tan2^-l). 

Prove  that  there  are  two  asymptotes 
perpendicular  to  the  polar  axis  at  dis- 
tances ±  a  from  the  pole. 


Part   IL 
THE   INTEGRAL   CALCULUS. 


CHAPTER   VI. 

TYPE    INTEGRABLE    FORMS. 

126.  Integral  and  Integration. 

If  f{x)  be  any  function,  and  f'(x)dx  its  differential,  then 
f{x)  is  called  the  integral  of  f\x)dx.  Hence,  any  function  is 
the  integral  of  its  differential. 

The  process  of  finding  the  function  from  its  differential  is 
called  integration.  As  an  operation  it  is  the  inverse  of  differ- 
entiation, and  having  seen 

I.  Differentiation  to  be  the  process  of  finding  the  ratio  of  the 
raies  of  change  of  the  function  and  its  variable,  we  may  define 

II.  Integration  to  be  the  process  of  finding  the  function  when 
the  ratio  of  its  rate  of  change  to  that  of  its  variable  is  given. 

127.  Symbol  of  integration.     The  symbol  of  integration  is 
I  ,  read  '  the  integral  of.'     Thus,  if 

y  =  ar\ 
then  dy  =  3  a^dx, 

and  i  dy  =  y  =  |  3  ay'dx  =  ■3?, 

d  and    |  ,  as  symbols  of  inverse  operations,  neutralizing  each 

other. 

The  test  of  the  result  of  any  integration  is  differentiation ; 

that  is,    I  ?>:i?dx  —  ar*  because  d(ar')  =  ?>v?dx. 

167 


168  THE   1NT?:(IHAL   CALCULUS. 

128.    Constant  of  integ^ration. 

It  is  evident  that  functions  which  have  the  same  rate,  and 
therefore  the  same  differential,  may  differ  from  each  other  by 
any  constant,  but  only  by  a  constant.  Thus  in  the  function 
y  =mx  +  &,  which  for  different  values  of  b  represents  a  series 
of  parallel  straight  lines,  the  rate  of  y  will  be  the  same  whatever 
the  value  of  b,  or  cly  =  mdx  for  all  values  of  b.  Hence  any 
given  differential  is  the  differential  of  an  infinite  number  of 
functions  which  differ  from  each  other  by  a  constant,  and  if 
the  differential  only  is  known,  the  function  cannot  be  deter- 
mined.    Therefore 

mdx  =  mx  -f-  C, 


/« 


in  which  C  is  an  undetermined  constant. 

Otherwise  :  since  the  differential  of  a  constant  is  zero,  if  a 
function  contains  any  constant  term,  this  term  will  not  appear 
in  its  differential ;  hence  a  constant  C  must  be  added  to  every 
integral  to  represent  this  term. 

This  constant  is  called  the  constant  of  integration.  It  will 
be  shown,  in  the  application  of  integration  to  definite  prob- 
lems, that  it  may  either  be  eliminated  or  that  its  value  may 
be  determined  from  the  conditions. 

129.  The  integral  of  the  sum  of  any  number  of  terms  is  the 
sum  of  the  integrals  of  the  terms. 

This  is  an  obvious  consequence  of  the  proposition  (Art.  16) 
that  the  differential  of  the  sum  of  any  number  of  terms  is  the 
sum  of  their  differentials. 

Or,  formally,  as   |   and  d  neutralize  each  other, 
d{x  —  y  +  z)  =  x  —  y-\-z  +  C, 
and  i  dx—  j  dy  +  i  dz  —  x  —  y  +  z  +  C; 


/" 


TYPE  INTEGRABLE   FORMS.  169 

hence  |  d(x  —  y-\-z)=  i  dx  +  i  dy  -\-  i  dz. 

Both  d  and   |   are,  therefore,  distributive  symbols. 

130.  If  the  differential  has  a  constant  factor,  its  integral  icill 
have  the  same  constant  factoi: 

For  dlAf{x)^=  Adlf{x)l  (1) 

and  fcJ[AtX^) ]  =  Af{^) ;  (2) 

but  (2)  is  the  integral  of  (1). 

Since  a  constant  factor  in  the  differential  also  appears  in 
the  integral,  siich  a  factor  may  he  loritten  before  or  after  the 
integral  sign,  at  pleasure. 

Thus,  d{ax)  =  adx,  and 


i  adx  =  a  I  dx  =  ax. 


131.  Type  integrable  forms. 

Since  a  function   is    the   integral   of   its  differential,  from 
d{ax"*)  =  max'^'^dx,  we  have 


I  max"'~\]x  =  ax"*, 

/__, ,        ax'" 
ax""  ^dx  = 
m 

Putting  m  —  1  =  n,  we  have  in  general, 

/ax^'dx  =  — ^  a;»+'  +  C. 
71  +  1 


71  +  1 

Reversing  the  fundamental  processes  of  differentiation,  we 
obtain  thus  the  twenty  following  forms  : 


1.  faa;"dr  =  — ^a;"+i  +  C'. 

J  7?  +  1 

2.  r^=loga:+C. 
J    X 


170  THE  INTEGRAL  CALCULUS. 

3.  I  a'  log  adx  =  a'  +  C. 

4.  I  e'dx  =  e'  -\-C. 

5.  I  cos  xrix  =  sin  x  +  C. 

6.  I  —  sin  xdx  =  oos  a;  +  C. 

7.  I  sec^  a;c7a;  =  tan  x  +  C. 

8.  I  —  eoseo^  ifdi'  =  cot  x-\-C. 

9.  I  sec  X  tan  a-r/iv  =  see  a;  +  C. 

10.  I  —  cosec  X  cot  a;c/.t*  =  cosec  x  -\-  C. 

11.  I  sin  a^a;  =  vers  x-\-C. 

12.  j  —  cos  xdx  =  covers  .r  +  C. 
1.3.  r     ^^'       =sin-^T  +  <7. 

14.  f -J£— =  cos-'.a-4-<7. 

J        Vl  -  ar' 

15.  r^^  =  tan-ia;  +  C. 

16.  r_^^  =  eot-^T-HC. 

17.  r ^^g =  sec-^  X  +  g 

*^  xVa*^  —  1 

18.  I ^1::::^:-^  =  cosee^  a;  +  C 

^       a;  Var^  —  1 


TYPE   INTEGKABLE   FORMS.  171 

19. 


^  V'^x-x' 

^^^^  =  covers"^  x  -\-  C. 

V2  X  —  ar 


20 


132.   Remarks  on  the  type  forms. 

The  processes  of  the  Integral  Calculus  consist  chiefly  in  the 
reduction  of  differentials  to  the  above  forms.  When  this 
reduction  has  been  effected,  the  integral  is  seen  at  once  by 
inspection.  This  being  the  case,  it  is  evidently  indispensable 
that  the  student  should  be  thoroughly  familiar  with  the  type 
forms,  so  as  to  be  able  to  recognize  them  at  sight.  The  fol- 
lowing suggestions  will  facilitate  their  recognition  and  appli- 
cation. 

Form  1.  Wlienever  a  differential  can  be  resolved  into  three 
factors,  viz. :  a  constant  factor,  a  variable  factor  vnth  any  constant 
exponent  except  —  1,  and  a  differential  factor  which  is  the  differ- 
ential of  the  variable  factor  ivithojtt  its  exponent,  then  its  integral 
is  the  product  of  the  constant  factor  into  the  variable  factor  with 
its  exponent  increased  by  1,  divided  by  the  new  exponent. 

For  •      Ca-x-.  dx  =  -^^  x"+'  +  C. 

J  n  +  l 

Form  2.  When  the  exponent  of  the  variable  factor  is  —  1, 
the  differential  falls  under  the  second  form 


f 


^  =  loga-  +  C, 


in  which  the  numerator  is  the  differential  of  the  denominator. 
Hence,  tvhenever  the  numerator  of  a  fraction  is  the  differential 
of  its  denominator,  the  integral  of  the  fraction  is  the  Naperian 
logarithm  of  its  denominator. 

Forms  3  axd  4.     These  forms  are 

a' '  log  adx  =  a"  +  C, 


f" 


172  THE  INTEGRAL  CALCULUS. 


and 


Ce'  •  dx  =  e'-\-C, 


in  which  the  differential  facto)'  must  be  the  logarithm  of  the  base 
into  the  differential  of  the  exponent. 

Forms  5-12.     In  eacli  of  these  forms  the  differential  factor 
dx  must  be  the  differential  of  the  arc. 

Forms  13-20.    The  conditions  to  which  the  differential  must 
conform  should  in  each  case  be  carefully  noted.     Thus,  from 


r-^?^  =  tan^..+ 
J  1  +  .t"^ 


a 


we  see  that  the  first  term  of  the  denominator  must  be  1,  and 
the  numerator  the  differential  of  the  square  root  of  the  second 
term  of  the  denominator. 

Examples.     1 .    |  ocp'dx  =  I  5  •  ar'  •  dr  =  |ar  +  C.     (Form  1.) 

2.  Cmx"'dx=-^^^  x'"--^C. 
J  1  —  m 

,y     Cadx       C     -2,1  ^^     I  ri 

3.  I  — —  =  I  ax  hJx  =  —  •-—+  C. 

J    X       a'  'Ix- 

.      r2dx         'i    ,  ri 
'•j35  =  -^+^- 

-     rdx^^2Vx-\-a 

6.  Cfax" -^^  +  Vx\ f7.T  =  V  -  —  +  2  J  ^ c.     (Art.  12^.) 

7.  Cb{a+bxydx=  C{a  +  bxy  ■  bdx=l(a+hxy  +  C.  (Form  1.) 

,^     8.     f    ^^^^^'     =  ^(4  +  ar^)-^3ar'f?.^^  =  2(4  +  :^-)^  +a 
^         ^   (4  +  .r')*     ^ 

9.    fm  (3  ao^  +  5  a^)  ^  (6  ax  4-  25  x*)dx  =  f  w  (3  ax' + 5  x")  ^  -j-  C. 


TYPE    INTEGRABLE   FORMS.  173 


r?J^  =  log(x^+l)^C.     (Form  2.) 
J  X-  +  1 


10 

+ 


In  logarithmic  integrals  it  is  customary  to  write  the 
constant  of  integration  C  =  log  c.     Hence 
log(a~'  +  1)  +C=  log(cB2  ^  1)  +  logc  =  log[c(ar'  +  1)]. 

.     11.    r_^  =  log[c(a;±a)]. 
J  X  ±a 

J  2  2  +  3a; +  ar'  2^2  + 3a; -far' 

.  =log[c(2  +  3ar  +  cr')^]. 

14.    ri-±^2i^da;  =  logrc(a;  +  sin.'«)]. 
.■'  X  4-  sin  X 

dx 

^^-  f^x  =fhk^  =  ^"^^^"^ ^>  +  ^^^ ^  =  ^""^^^ ^"^^-l- 

16.  flOlog'a;  — =  flog^a;-fa     (Form  1.) 

J  X        "  I 

17.  f m  log»a;  —  =  ^^  log»+'a-  +  0. 
c/  a;       n  -|- 1 

18.  J  ae'^clx  =  |  e"  •  adx  =  p"'  +  C     (Form  4. ) 

\j  19.  r3  log  aa'Vda;  =  fa''  •  log  a  3  a^dx  =  a^  -f  C.     (Form  3. ) 

20.  r e"'"  Vos  a-da;  =  e"'"  "^  +  C. 

21.  I  sin  X  cos  xfix*  =  \  sin-  a*  +  €'.     (Form  1.) 

22.  f-  2  sin  2  .rda-  =  f-  sin  2  x  •  2  dx  =  cos  2  a;  +  C. 


174  THE  INTEGRAL   CALCULUS. 

23.  J  4  sin'^  x  cos  xdx  =  sin*  x  +  C. 

24.  I  4  sec* a;  tan  xdx  =  I  4  •  sec'*  x  •  sec  x  tan  ccdx  =  sec*  x  +  C 

25.  I  ^ tan* a; sec'  xdx  =  J^  ta n'  x-\-C. 

26.  I  6  tan  ar*  sec^  a:''  •  .T^dr  =  I  2  •  tan  x^  sec"  a;*  •  3  a:?dx 

='tan2ar^  +  C. 

27.  C-^^^  =  sin-'x'  +  C. 
J  Vl  -  a;* 

oo     /^2sin"^^a^a;       /^o      •   -i  <^^  /  •   -i    xo  ■  /-» 

28.  I  —  =  I  2  •  sm  ^a; =(sin  ■'a;)-+(7. 

(Form  1.) 

29.  I —  -  =  I —  =  vers^oa;  +C. 
-'  VlOa;  -  25a^'     •'  V2(5a;)  -  (5.^;)- 


V2(5a;)-(5.'c)-' 

dx 

+  4a;  +  5     J  l+(a;-2)' 


30.    C-—-^ =  f- ~ =  tan-^a;  -  2)  +  C. 

Jx'  +  ^x  +  o     Jl+(x-2V  ^  ' 


31. 


Ce'^e'dx  =  ("^+C. 


TiTiTnVTF.NTARY   TRANSFORMATIONS. 

No  general  method  exists  for  the  reduction  of  differentials 
to  type  forms.  Much  therefore  depends  upon  the  ingenuity 
and  insight  of  the  student.  In  addition  to  the  specific  trans- 
formations applicable  to  certain  differentials  of  definite  forms, 
given  in  the  next  chapter,  the  following  elementary  transforma- 
tions should  constantly  be  borne  in  mind. 

133.   By  the  introduction  of  a  constant  factor.     When  the 

differential  is  under  a  type  form  so  far  «.s-  the  variable  is  con- 
cerned, it  may  frequently  be  reduced  exactly  to  such  form  by 
multiplying  and  dividing  by  a  constant  factor.     This  reduction 


ELEMENTARY  TRANSFORMATIONS.        175 

depends  upon  the  fact  that  a  constant  factor  may  be  written 
before  or  after  the  integral  sign. 

Illustrations.      Fo'rm  1.        |  {3aa:r-\- 2xy  -  {3ax -\-l)dx. 

Were  the  differential  factor  (6ax-\-2)dx,  it  would  be  the 
exact  differential  of  the  variable  factor  without  its  exponent. 
Hence,  multiplying  and  dividing  by  2, 

CiSax"  +  2xy(Sax  +  l)rlx  =  J-  C(3ax-  +  2xy{6ax -f  2)dx 

=  j^{3ax'  +  2xy-]-C. 

When  the  proper  factor  is  not  readily  seen  by  inspection, 
we  may  determine  it  as  follows.  Suppose  the  differential  to 
be  (2a;2-|-a-5)'f(6a;?  _|_2ic^)dir,  and  ^  =  required  constant  fac- 
tor.    Then  A  must  satisfy  the  condition 

d(2x^  +  «*)  =  (^Ax^  +  2Ax*)dx, 

or  {3x^--\-5x*)dx  =  {^Ax^  +  2  Ax*)  dx ; 

and  as  this  condition  must  be  fulfilled  for  all  values  of  x,  the 
coefficients  of  like  powers  in  the  two  members  must  be  equal, 
or  ^A  =  3,  2A  =  5,  from  either  of  which  we  find  -<4  =  f.  In- 
troducing this  factor, 

iC(2x^  +  x^)'(3x^'-\-5a^)dx  =  -^{2x^  +  x^)'i+a 

Again,  suppose  the  given  differential  to  be 

(2a^  +  7x)^(5a^-\-3)dx. 

Then  we  must  have 

d(2ar'  +  7x)  =  (6a^  -f  7)dx  =  {oAx?  +  3A)dx; 

whence  6  =  5^,  and  1  =  3 A,  or  J.  =  |,  ^=^.  As  these 
values  are  not  the  same,  there  is  no  constant  factor,  and  the 
integration  cannot  be  effected  by  Form  1. 


176         thp:  integral  calculus. 


Form  2.  C^+M 
J  Qx  +  x'' 


dx. 

ox-\-x'* 

Were  the  numerator  6  +  4  ar\  it  would  be  the  exact  differen- 
tial of  the  denominator.     Hence,  introducing  the  factor  2, 

I  a     ,  ^^^  =  ^\  i,     ,     ,=  i^og(6x  +  x*)  +  logc 
=  \og[c{(^x+x'y-]. 

If  the  constant  factor  is  not  readily  seen,  it  may  be  deter- 
mined from  the  condition  that  the  numerator  must  be  the 
exact  differential  of  the  denominator. 

FoBMS  3  AND  4.  The  constant  is  determined  from  the  con- 
dition that  the  differential  factor  must  be  the  product  of  the 
logarithm  of  the  base  into  the  differential  of  the  exponent. 
Thus,  to  integrate  a^dx,  the  factor  to  be  introduced  is  2  log  a, 
and 

Ca^dx  =  — —  fa^  ■loga2dx  =  —^ —  a^  +  C. 
J  2  log  a  J  2  log  a 

Forms  5  to  12.  The  required  constant  is  readily  seen  from 
the  fact  that  the  differential  factor  is  the  differential  of  the 
arc.     Thus 

I  cos 2xdx  =  Y  I  cos  2x-2dx  =  ^sm2x-{-  C. 

Forms  13  to  20.  In  the  case  of  the  circular  differentials 
the  constant  must  be  determined   separately  for  each  form. 

/dx 
—  ,  we  observe  that  so  far 

as  the  variable  is  concerned  it  has  the  type  form    I  —  • 

To  transform  it,  we  must  make  the  first  term  under  the  radi- 
cal 1,  and  the  numerator  the  exact  differential  of  the  square 
root  of  the  second  term  under  the  radical.  We  proceed,  there- 
fore, as  follows : 


ELEMENTARY   TRANSFORMATIONS.  177 


1  h 

-  dx  -,     _      -dx 


\        a-  \        a- 


-x" 


=  -  sin  ^-x  4-  C/- 
6  a 


(14-)     r ^^        =-cos~'-a;+a 

1 


1  6 


=  1  tan  1  -  a;  +  C. 
cib  a 


(16')    f-  -:r^—  =  \  cot-^i  ^  x-  +  a 
J       a-  -\-  b-xr      ab  a 

h 
(17')    f        ^^"'         =^    f_Jf==l  f ^'^'' 


1         ,  ^ 

=  -  sec~*  -  x-\-  C. 
a  a 


.   ,      r  dx  1  ,  b  „ 

(18')    I ,  =  -  cosec-i  ~x  +  a 


1  6 

dx'  ^    _         -  da; 


-V  —  a; .;  XT  \\ —  X 7,x 

\  a         a-  \  a  a^ 


1         -lb      ,   rt 
=  -  vers  ^  -  X  +  C  • 


6  a 


do;  1  ,b 


J  ax  L  1  "  ^ 
-  =  V  covers-^     x  +  C. 
V2«6x-6V      &                 a 


178  THE  INTEGRAL  CALCULUS. 

These  eight  forms  are  known  as  the  subordinate,  or  auxiliary, 
circular  forms.  It  is  better  to  transform  each  special  case 
directly;  thus 

4  V3  C  Vffte 


r  Adx    ^4  r    (ix    ^4V3  r 

J  'S-\-  i)x-     3  J  1  +  ix"^     3  -y/BJ  1 


^     tan-V|a;  +  a 


V15 

If  the  subordinate  forms  are  memorized,  or  at  hand  for  refer- 
ence, we  have 

4:dx     _      dx 
3  4- oar'      f-ffar' 

whence,  by  comparison  with  (15'), 

a^  =  I,    //'  =  ^, 
and  hence 

1  .       ,  ^         ..        4 

-T  tan  '  ~  x-\-  C  =  —r— 
ah  a  -\/lh 


tan  ^  ~x-\-  C=  -7==  tan  '  Vfic  +  C 


Examples. 


y      1.   J|(.r'  +  l)'^a^da;=|4.J(af+l)WdT=^\(af  +  l)^  +  C. 

4.  rV2a;^-3a:^  +  l(.t''  -  f  a;)da;  =  tV(2cc*  -  3.t2  +  1)2  +  C. 

5.  Which  of  the  following  can  be  integrated  by  introducing 
a  constant  factor  ? 

{o^  4-  Sar'  +  a;-  +  5)^(2 x'  +  ^ar'  +  ^a;)da;. 

(3a^-2x)'(3a;-l)da;. 

3.'r— 1        ,          4-f6a;-      , 
■dx.   ■ ~  Ax. 


(l-aj  +  .r^)-  (4a;-3ar')^ 


ELEMENTARY  TRANSFORMATIONS.        179 

^    a,v^  —  bx^-  "^^     ax"  —  hx- 

=  log  [c  (aar*  —  6.x'- )  *]. 

7.     f-    ^^^•^'    =log[c(12ar^  +  7)^'n. 
J  412aT^  +  7         ^'-  ^         -1-    /    J 

J  a  —  o.r* 


1 
(a  —  6ar')".» 


9.     f    ^^"'^•^'     =logrc(10ar'4-16)^]. 


+ 
10 


/'   sin  xdx    _  , 
a  +  6  cos  a; 


+  6  cos  X         "  /     ,   t  V  r 

11.    Which  of  the  following  can  be  integrated  by  the  intro- 
duction of  a  constant  factor? 

odx          1— V■^'7         x'"~^dx       Ax—'S^x-, 
-dx. ax. 


8-6ar^       ^_j^  x^  +  l  .^_J 

2 .  I  (("dx  = a"'  +  0. 

J  a  log  a 

3.  (a^'xHx  =  —1—  a-'  +  G. 
J  3  log  a 

4.  I  nifi^^dx  —  #?«e'-'  4-C. 

5.  Te'" ' sin .xdit  =  -  e™' '  +  C 
<J-   J  e     2cos^^f7a;  =  2e    -'+C. 

1  +  or 

18.    Which  of  the  following  can  be  integrated  by  introducing 

a  constant  factor  ? 

"  / .  t 

(f  a  loa  a  —        e-*^  ndx.     e'dx. 


180  THE   INTEGRAL   CALCULUS. 

19.     I  CDS'*  X  sill  xdx  =  —  I  cos^  x  -\-C. 


20 
21 
22 
23 


.     I  sin-  4  X  cos  4  xdx  =  jij  sin'  4  a;  +  C 

I  sin  a^  •  2(?dx  =  —\  cos  af'  +  C 

I  f  sec^  0/-^  tan  x^  •  x-rfo;  =  \  sec^  ar'  +  C 

3.     f       ^^        =-Vsin^#x-+C. 
J  V4-9ar 

24.  f _^£__  =  J_cos-i-  +  C. 

^       Va(62_ar')      Va  & 

25.  f-A^  =  2tan  i2a;  +  C. 
J  1  +4af 

26.  f-^^^ltaii  laj^+C. 
J  1  +ar 

27.  f,;l^  =  _^tan-'Via^+C. 
J  o+  ix"      V35 

--gji =  -Jl-sec-'  Vfa;  +  (7. 

xVSa:^  — i")      Vo 

/da;  _,a^  ,  ^ 

— — — -— ^^  =  vers     -  -\-C. 
V2  a.r  —  .r-  <<■ 


29 


134.  By  the  transference  of  a  variable  factor.  Although  a 
variable  factor  cannot  be  taken  out  from  under  the  integral 
sign,  it  may  be  transferred  from  one  factor  of  the  differential 
expression  to  another,  or  introduced  into  both  terms  of  a  frac- 
tion. 

Illustbatiox. 

ff  (ax"  +  a;«)^(5a  +  7ar')da;  =  f  C(aaf  +  x-)^(5ax*+7afi)dx 

=  (aar*  +  .f')'+C. 


ELEMENTARY   TRANSFORMATIONS.  181 

Examples. 
1.    I  —  -dx=  I — =^=dx  =■  2Xct^  +  xy  +  C. 

"  -^  (1  -  o^)  I  "-^  (a;-2  _  1)  I  ~  VTZ^ 
dx 

g    r_jodx__  _  C__^__  _  _  C(^    1  V¥_  ^^ 


5  r    ^jg    _  c ^ 1  c 


\        a;* 

1  .     .a      ^ 

= sin-i-+C. 

a  a; 


•^  ^aj^l+a;^)     *^  l4-.r^ 


135.  By  expansion.  When  the  exponents  of  the  factors  of 
the  differential  are  positive  integers,  the  indicated  operations 
may  be  performed,  and  the  resulting  monomial  terms  inte- 
grated separately.  Care  should  be  taken  not  to  expand  un- 
necessarily ;  thus, 

J(l  -  xydx  =  -  i(l  -  xy^  4-  C. 
Examples. 


182  THE   INTEGRAL   CALCULUB. 

2.  Jix  +  lya^dx  =  I'  +  |a^  + 1 a.4  ^  ^_^  _^ C'^ 

3.  C(a  +  hxfxdx  =  '^x-  +  '^x'  +  -V  +  C. 

*J  ^  O  4: 

4.  C(l  -x  +  x^ydx  =  x-x^  +  x'  —  :^x^+\x''+C. 

5.  I  ^(1 +  sin4a;)c/a;  =  ^(aj  — ^cos4a;)+C. 

J  3  7  11  lo         " 

136.   By  division.     Expansion  by  division  will  often  lead  to 
integration,  as  may  be  seen  by  the  following 

Examples. 

2-    r?^c?x  =  ix*-^ar'  +  log(l+.'r')+C. 

3.  C^^±ldx  =  ^x''  +  x  +  ]og{x-iy+C. 

4.  r(^  +  ^^ydx  =  9  log  x-  +  VWx  +  X  +  C. 


137.  By  separation  into  partial  fractions  having  a  common 
denominator, 

Since        m+*S-la,=§^a.+^^d., 

a  fraction  may  be  separated  into  partial  fractions  having  a 
common  denominator,  and  thus  integrated,  if  the  partial  frac- 
tions are  integrable. 


ELEME^'TARY   TRANSFORMATIONS.  183 

Examples. 
1.    f?Lt^d^=fj^+fi^ 

b 

=  a  tan~'  x  +  log(l  +  x^)-  -f-  C. 


2-    fx/^-^  dx  =  r-A±^da;  =  sin^»  a:  -  (1  -  ar')  -*  +  C. 
J  M-x         -'Vl-ar^ 

Wa^-1)^ 
3.    I dx  =  {x- —  1)- —sec^x-^C. 


4.    r_^^xdx^  ^  r 


a  —  2  a;  —  a  , 
ax 


■y/ax  —  a?     ••'   2 Vwa;  —  xr 

/'   g  —  2a;     j._ft/^      c?a; 


2 Vaa;  —  ar*  ^»-'  Vaa;  —  ^ 

=.{ax  —  x?Y  —^\QX^^-x+C. 


a 


J  ar'+4a;     4J    ar^  +  4a;  4J  ar'+4a;      4J  ar'+4a; 

Ux  +  A      Ux  L\^  +  vJ 


CHAPTER   VII. 
GENERAL    METHODS    OF   REDUCTION. 
BY    PARTIAL    FRACTIONS. 
138.   Rational  Fractions.     Every  fraction  of  the  form 

a'x"  -I-  6 '.T"-'  H Vx-\-'k'' 

in  which  m  and  n  are  positive  integers,  is  called  a  rational 
fraction. 

It  is  evident  that  every  such  fraction  can  be  reduced  by 
division  to  a  series  of  monomial  terms  plus  a  rational  fraction 
Avhose  numerator  is  of  a  lower  degree  than  its  denominator. 
Thus, 

^"  a.-  + .«"  -  1  H-  -./  ~  ^ 


x^  —  x  +  1  ar  —  x-\-l 

As  the  monomial  terms  can  be  integrated,  we  are  concerned 
only  with  rational  fractions  whose  numerators  are  of  a  loAver 
degree  than  their  denominators,  and  we  are  to  show,  — 

1°.  That  every  such  fraction  can  be  resolved  into  partial 
fractions  whose  denominators  are  factors  of  the  denominator 
of  the  given  fraction,  and 

2°.  That  these  partial  fractions  can  always  be  integrated. 
There  will  be  four  cases,  according  as  the  factors  of  the  de- 
nominator of  the  given  fraction  are 

1.  real  and  unequal,  3.  imaginary  and  unequal, 

2.  real  and  equal,  4.  imaginary  and  equal. 

184 


EEDUCTIOX   BY   PAKTIAL   FRACTIONS.  185 

139.   Case  1.     The  factors  real  and  unequal. 

1°.  Let  •'^   '  be  the  fraction,  ^(x")  being  resolvable  into  n 

real  and  unequal  factors  ic  —  a,  x  —  6,  •••  x  —  n.    Then,  to  every 
factor  x  —  a,x  —  b,---x  —  n,  tliere  corresponds  a  partial  fraction 

A  B  2^ 


x  —  a     X  —  b       X  —  11 
in  whicli  A,  B.  •••  N are  constants,  or 

f(x)  _     A      ^      B      ^  2^ 


<f)(x)      X  —  a      X  —  b 

It  is  required  that  this  equation  shall  be  an  identical  one, 
true  for  all  values  of  x.  Reducing  the  second  member  to  a 
common  denominator,  this  denominator  will  be,  by  hypothesis, 
<i>{x),  and  the  sum  of  the  numerators  will  be  equal  to  f{x). 
This  sum  will  be  a  polynomial  of  the  (h  —  l)th  degree,  and 
since  the  equation  formed  by  placing  it  equal  to  f{x)  must  be 
true  for  all  values  of  x,  the  coefficients  of  like  powers  of  x  must 
be  separately  equal.  We  shall  therefore  have  n  equations  of 
condition  from  which  to  find  the  values  of  the  n  constants 
A,  B,  '••  N.     Hence  the  resolution  can  always  be  effected. 

fix) 
2°.  The  integration  of  ^^  '  dx  is  thus  made  to  depend  upon 

jA  dcx* 

that  of  a  series  of  fractions  of  the  same  form,  namely 

/Adx  ■  x—a 
=  A  log  (x  —  a).     Hence  the  integration  is  always 
x  —  a 

possible. 

Examples. 

1.      ^^  +  ^     dec  =  5.T  -f  15  -I-  -l^^^:!?-^--  by  division. 

The  factors  of  x?  —  '6x-\-2  are  x  —  1  and  .t  —  2 ;  hence 

35a; -29    ^    A  B    ^  ^(o;- 2)  + -B(a;- 1) 

a^_3a:-|-2      x-1      a;-2  x'-'6x-^2         ' 


186 
whence 


THE   INTEGRAL   CALCULtTS. 


Sox  -  29  =  A{x  -  2)  +  B(x  -  1) 
=  {A-\-B)x-2A-B. 
Equating  the  coefficients  of  like  powers, 
3o  =  A  +  B,   29  =  2A  +  B; 
therefore  A  =  —  6,  B  =  41,  and 

Jx^-:^x-\-2         J  .;  J  x-l     J  x-'J 

=f  a^+laiK-f)  log(a;— 1)  +411og(a;-2)  +  C 

=  l^  +  i5a;  +  iog(;"-^)Va 
(x  —  ly 

2-    ^^Z^f'^S^^x.     The  roots  of   ar- -7.^2  +  36  =  0  are  G, 
or  —  (  ar  +  36 

3,  and  -  2. 


Hence   2a^-3x  +  5^^^ 


B 


ar»-7a^  +  36      x-6      x-'S      x-\-2' 
and  2x2  -  3 cc  +  -  ^  ^^^  _  3^  ^^  ^  o)  ^  ^^3,  _  6)  (a;  +  2) 

•      +C(a;-6)(a;-3). 

Instead  of  proceeding  as  in  Ex.  1,  the  vahies  of  the  con- 
stants are  readily  found  by  assuming  some  value  for  x,  since 
the  equation  is  to  be  true  for  all  values  of  x.  Thus,  making  x 
equal  to  —  2,  3,  and  6,  in  succession,  we  find  C=  \^,  B  =  — 1|-, 
A  =  ^,  and 


r2x^- 

J  a^- 


—3x+5 


7a;2  +  36 


dx  =  lo 


-6)^^  (a; +  2) 
a;-3)T^ 


3.    r4^da.  =  log j(-^::^c. 


REDUCTION   BY   PARTIAL   FRACTIONS. 


187 


6.  r^^=iog,p(^-^). 

Ja^-9         ^\    x  +  3 


x  +  h 


'•/. 
'•/. 


J^^^dx=.\og[c(x-^)Hx  +  2fy 


10 


a^  -{-x^  —  4:X  —  4 
3a^-l 


dx  =  log 


■^(x-^i)ix-2y 
ix+2y 


.     I  -^ dic  =  log  [o(.r  +  1)  (.r  -  l)x]  =  log  [c(.'r'  -  a;)]. 

"^    ^~^  (Form  2.) 


140.   Case  2.     TAe  factors  real  and  eqiml 

f(x) 
1°.    Let  -^^   ^  be  the  fraction,  <^(a;)  being  resolvable  into  n 
fi>(x) 

real  and  equal  factors  x  —  a,  a;  —  a,  •••.     Then,  to  such  set  of 

n  equal  factors  there  corresponds  a  set  of  n  partial  fractions, 

A  B  N 


(a;  — a)"'    (a:  — a)"  ^       x  —  a 
stants,  or  '^-^  = 


,  in  M-^hich  A,  B,  "•  N  are  con- 


+ 


B 


irn+- 


N 


<f>{x)       (x  —  a)"      (cc  — a)"~"  x  —  a 

Reducing  the  second  number  to  a  common  denominator, 
this  denominator  will  be  equal  to  <l>{x),  and  the  sum  of  the 
numerators  will  be  a  polynomial  of  the  (n  —  l)th  degree  equal 
to  f(x) .  The  latter  equation  is  to  be  an  identical  one  true  for 
all  values  of  x ;  hence,  equating  separately  the  coefficients  of 
the  like  powers  of  x,  we  have  n  equations  of  condition  from 
which  to  find  the  values  of  the  n  constants  A,  B,  •••  N.  The 
resolution  is,  therefore,  always  possible. 

When  the  factors  of  </>(a;)  are  not  all  equal,  the  two  cases 
can  be  combined.     Thus 


JM. 


A 


{x-2y{x-3y{x-4:)      ix-2) 

D 


7-,+ 


B 


+  ■ 


C 


+ 


-.+ 


{x-2y  '  x-2 
E  F 


{x-Sy      a; -3      x-4: 


188  THE  INTEGRAL  CALCULUS. 

2°.  The  integration  of  -^   ^  dx  is  thus  made  to  depend  upon 
<f>(x) 

that  of  a  series  of  fractions  of  the  form If  n  =  1, 

(a: -a)" 

rAdx^  =  A  log  {x  -  a).     If  n  is  other  than  1, 
J  X  —  a 

r^Mx_  ^ ^  r^^ _  . . „^^ ^ _A^  .^ _  .!.„ 

J  {x-ay         J  '  \-n  ' 

Hence  the  integration  is  always  possible. 

x  —  \ 
Examples.     1.  -da;. 

Placing 

a;-l    ^       A  B     ^^  +  -5(^  +  1) 

{x  +  \y      {x-ir\f     x^-\  (a;  +  l)-      ' 

and  equating  the  numerators,  we  have  a:  —  1  =  ^  -|-  Bx  +  B. 
Placing  the  coefficients  of  like  powers  equal,  we  obtain  B=\, 
A  =  —  2;  whence 

,       ,    1.2^^=        ,       .    -,,0+  I   — — r=""-r+log(a;  +  l)+a 

{x  +  \y      J  {x  +  iy   J  x-{-i    x-\-i 

2    (a;^  +  o)dx 

•   {x-iy{x  +  2){x  +  l) 
This  is  a  combination  of  Cases  1  and  2,  three  only  of  the 
factors  being  equal.     Hence  we  assume 

^  +  5  _       A  B  C 


(ar-l)''(a;  +  2)(x  +  l)       (a;-!)''      (a;-!)-      a;  - 1 


a;  +  2      X  +  1' 

whence,  reducing  to  a  common  denominator,  and  equating  the 
numerators, 

»*  +  5  =  ^(a;  +  2)  (a;  +  1)  +  B{x  -  1)  (a;  +  2)  (a;  +  1) 
^C{x-iy{x  +  2){x  +  l) 
+  D{x-iy(x  +  l)  +  E{x-iy{x  +  2),         (1) 


REDUCTIOX   BY   PARTIAL   FRACTIONS.  189 

=  (C  +  D  +  E)x*-{-{B-{-C-2D-E)x^ 
+  {A-\-2B-3C-SE)jf 
-^{3A-B-  C  +  2D  +  5E)x 
+  2A-2B  +  2C-D-2E.  (2) 

In  (1),  make  x  =  —  l,  x  =  —  2,  x  =  1,  in  succession,  and  we 
have  directly  E  =  —  ^,  D=^,  A=l.  Equating  the  coeffi- 
cients of  a;*  in  (2),  C-\-  D  +  E  =  1,  whence,  having  D  and  E, 
C=  If.     Equating  also  the  absolute  terms, 

o  =  2A-2B  +  2C-D-2E, 
whence  B  =  —  ^.     Therefore 

r         (x*  +  o)dx  _  r     dx      _  1  r     dx         35  r  dx 

J  {x-lY{x-\-2){x-\-l)~J  (x-iy     GJ  {x-iy     36J  x-1 
7  r  dx    _  3  r  dx    ^  _         1  1     1 

QJ  x-\-2      4 J  X-  +  1  2(a; -  1)2 "•" 6 a; -  1 

+  Mlog(^'-l)+ilog(^-f2)-flog(a^  +  l)  +  C. 

3.    C^^-'^  dx  = —  +  log{x-3y-\-C. 

J  {x-3y  x-3         ^^  ^ 

J  {x-iy{x-2)      x-1  ^x-1 

r      {2x-5)dx       ^ L_  +  iilog^+l4-C. 

J  af'_|_5r'  +  7a;  +  3      2(a--M)       ^      ^x-\-3 

'■  /(.-2r;+3)-=-2T,G4-2+^3)+^^-^^^^- 
r     ix  +  2)dx     ^U_l 3_\     ^^-1 

J  {x-iy(x+i)    ^\x-i    (x-iyy^  ^x+i^ 
J  {x-3y       {x-3y 


141.   When  the  factors  of  <}>(x)  are  imaginary,  the  above 
processes  will  lead  to  the  logarithms  of  imaginary  quantities. 


190  THE  INTEGRAL  CALCULUS, 

To  avoid  such  we  resolve  <f>{x)  into  quadratic,  instead  of  sim- 
ple, factors,  as  follows  : 

The  general  form  of  an  imaginary  quantity  being  a-|-6V  — 1, 
that  of  an  imaginary  factor  will  be  aj  —  (a  +  ftV  — 1).  But  for 
every  such  factor  there  must  be  another,  a;  — (a  — 6V— 1), 
since  <l>(x)  is  real.  Therefore,  for  every  pair  of  imaginary 
factors,  ft>{x)  will  have  a  quadratic  factor  of  the  form 

[a;  _  (a  +  &V^][.x  -  (a  -  6  V^)]=  {x  -  a)-  +  &'• 

Cash  3.      The  factors  imagiyiary  and  unequal. 

f(x\ 
1°.  Let  -'^   '  be  the  fraction,  ^(x)  being  resolvable  into  p 
<l>{x) 

unequal  quadratic  factors  (x  —  a)^  +•  ^^  (^*  —  ^Y  +  d^,  etc. 
Then,  to  every  such  quadratic  factor  there  corresponds  a  par- 
tial fraction ^t_^ -,  "^ -,  etc.,  in  which  A,  B, 

(^x-af  +  h''    {x-cy  +  d^ 

C,  D,  etc.,  are  constants,  or 

f{x)  ^      A  +  Bx  C+Dx  M+  Nx 

<l>{x)  ~  (^x-ay  +  b'      {x-cy  +  d'  {x-my  +  n^' 

For,  in  reducing  the  second  member  to  the  common  denomi- 
nator <f>(x),  any  numerator,  as  A-\-Bx,  will  be  multiplied  by 
p  —  1  factors  of  the  form  (a;  — a)- +  6',  and  the  sum  of  the 
numerators  [=/(ic)]  will  therefore  be  a  polynomial  of  the 
[2(p  —  1)  +  l]th  degree.     We  shall  therefore  have 

2(p_l)  +  2  =  2p 

equations  of  condition  from  which  to  find  the  values  of  the 
2p  constants  A,  B,  '••  N.  and  the  resolution  is  always  possible. 

2°.  The  integration  of  -'^  ^  dx  is  thus  made  to  depend  upon 
<f>(x) 

that  of  a  series  of  fractions  of  the  form 

(A-^-Bx)dx  _  {A  +  Ba)dx        B(x  —  a)dx 
{x -  ay  +  b^~  {X-  ay  +  b-      {x  -  ay  +  b^' 


REDUCTION    BY   PAKTFAL   FRACTIONS.  191 

r{A  +  Ba)dx  ^  A±Ba^^^_,  x-a  .^^.^  ^33  ^^^,.. 

^  rB{x-a)dx       B^^^^^_      ,       ,^^ 

Hence  the  integration  is  always  possible. 

Examples.     1.  -^- ~    / 

a;^  —  4  a;  +  5 

The  factors  of  ic^  — 4a;  +  o  are  x  —  {2±  V— 1),  and  their 
product  is  (a;  —  2)-+l,  or  a  =  2,  6=1  in  the  form  (x  —  a)^-f-  6*. 

Assuming  — — ^^^^ = — — - — ,  we  have  A=  —4,  B=l. 

^a^^ix  +  r>      (a; -2)2  +  1'  ' 

Hence 

/(x  —  4:)dx       A-i-Ba.       .x  —  a.B-,      r-,  \'>  ,  ^,i-^ 

x^—  Ax-\-  i^  b  b         2 

=  -  2  tan-H.r  -  2)  +  ^log  [(x  -  2)^  +  1]  +  C. 
f,    (a;^  +  a:^  +  a;  +  1 )  da; 

(.T-l)*(.T-'  +  2) 

Assume  -^^±^i±i^±l- =  -^A- + -A_  +  £±^, 
(a;-l)2(a^  +  2)       {;x-\y      x-\        x'-\-2 

whence  A  =  ^,  5  =  Y-,  ^'  =  |»  D=  —  h  ^"'^ 

J(ar'  +  ar^  +  a;  +  l)da;^4  r     da;  10   r  dx 

{x-iy{x'  +  2)     "sJ  (a;-l)-       9  J  a;-l 

5  r  dx    _  1   r  xdx 
9Ja:2_^2      9Ja;2  +  2 

4     1       ,  IOt      ,        1, 

s= log  (a;  —1) 

3a;-l       9      ^  ^ 

4.  _A^  tan-i  _^  - -^  log  (ar^  +  2) -f  C. 

9V2  V2  -   18 

„  ar^da;  _  ^da;        5da;       (C  +  Dx)dx 


(a;  +  l)(a;-l)(a-2  +  2)      x  +  1      a;  - 1  a;- +  2 


192  THE  INTEGRAL  CALCULUS. 

Then 
a.^  =  A(x-l)  (a^  +  2)  +  B(x  +  1)  (ar'  +  2) 

+  (C+Dx)(x-\-l){x-l)  (!) 

=  (A  +  B-\-D)a-f'  +  (B-A  +  C)ar 

+  {2A  +  2B-D)x  +  2B-2A-C.     (2) 

From    (1),   when   x  =  \,    x  =  —  l,    in    succession,    we   have 
B  =  \,  A  =  -\. 

From  (2),  B-A  +  C=l,  or  C  =  f ;   and  ^  +  5  +  7)  =  0, 
or  Z)  =  0. 

Therefore 

r ^da; ^  _  1  r  dx        1  r  dx        2  r    x 

J  {x-{-l){x-  1) (x^  +  2)  qJ  x  +  1      6J  x-1      sJ  ar'+2 

11      a;  —  1  ,   V2 ,      -1    X     ,  ^ 
=  ilog— — +— -tan  '-— +  C. 
a;  -(- 1        3  V2 

da; 


J  (x-l 


L-tan-'-^+a 

3V2  V2 


+  ^tan-*a;+C. 

142.   Case  4.     The  factors  imaginary  and  equal. 

1°.   Let  <l>{x)  be  the  fraction,  ^'^   ^  being  resolvable  into  p 

<f>{x) 

equal  quadratic  factors  (x  —  a)^+  6^,  (a;  —  a)^+  b^,  etc.     Then, 

to  such  set  of  factors  there  corresponds  a  set  of  p  partial 

fractions, 

A  +  Bx  C  +  Dx  M+Nx 

\_{x  -  ay  +  6-]'''    [(a-  -  ay  +  Wy-''  '"  {X-  ay  +  W' 

in  which  A,  B,  •••  N  are  constants,  or 

f{x)  ^        A  +  Bx  C+Dx  M+Nx 

4,{x)      {{x-ay  +  Wy     {{x-ay  +  U'Y  ''^"'  Xx-ay  +  i/ 


EEDUCTIOX   BY   PARTIAL   FRACTIONS.  193 

Reducing  the  second  member  to  the  common  denominator 
<l>{x),  the  sum  of  the  numerators  [  =  /(ic)]  will  be  a  poly- 
nomial of  the  [2(i>  -  1)  +  l]th,  or  (2iJ  —  l)th,  degree.  This 
equation  will  therefore  furnish  2p  equations  of  condition,  from 
which  the  values  of  the  2p  constants  can  always  be  determined. 
^The  resolution  is,  therefore,  always  possible. 

2°.  The  integration  of  -^^^  dx  is  thus  made  to  depend  upon 
<t>{x) 

that  of  the  general  form  — ^^ — — ^— — ,  in  tvhich  jy  is  integral. 

\_{x  —  a)^-f  6^]'' 

If  2^  =  Ij  tlie  integration  has  been  shown   to   be   possible  in 

Art.  141.      If  p   is   other   than   1,    place   x  —  a  =  z,   whence 

x  =  z  +  a,  dx=  dz. 

Then 


/ 


{A-\-Bx)dx 

/{A -\- Bz  +  Ba)dz ^  r    Bzdz  r{A  +  Ba)dz 

{z^  +  l^y         "J  (z^ +&')".'     (z'  +  ft^)" 


2(p-l)(22-f-62)» 
be   shown   in  Art 
is  always  possible  when  p  is  integral. 


and  it   will    be   shown   in  Art.  147,  that  the  integration   of 
dz 


(z*  +  by 

Examples. 

^     r{7?-ifx'-\-2)dx^  r{A  +  Bx)dx      r{C-\-Dx)dx 

whence  af  -\-  .i-  -\-  2  =  A  +  Bx  +  (C  +  Dx)  (ar  +  2),  from  which 
we  find  ^  =  0,  B= -2,  C=D  =  1.     Hence 


r(x^-{-jr  +  2)dx  ^  r  -2xdx       r   dx         f 
J         {ii^  +  2Y  J  {x"  +  2y     J  x'  +  2     J  a 


xdx 


x2  +  2 
1     +_i_tan~^-i^+ilog(x^'+2)+C. 


194  THE  INTEGRAL  CALCULUS. 

,,     r(x*+2x^—2jf—2x+5)dx  ^  A+Bx      C+Dx        E 
'J  (ar'  +  l)=^(a;-2)        '      {x^+\y       x^'+l       x-2' 

whence  ^1  =  —  4,  .5  =  0,   C  =  2.  D  =  0,  E  =1,  and  we  have 

r  —-idx         r  2dx        r  dx 
J  (or^  +  1)-     J  x^  +  1     J  X  -  2 

=  2  tan-^  X  4-  log  (a;  -  2)  -  4  f — — 


3     ri^-x  +  l)dx  ^^      (x  +  iy  _^tan-'.>;  + 

4.     p^  +  -»^  +  x-'  +  .^-   ^,^ o 

J  r.r  +  2)Hx'  4-  .S)^  2(a^  +  2) 


a 

10 


{x'  +  2y{x'  +  l^y  2(a^  +  2)      x'  +  'S 

+  ^log(r'+2)-91og(ar+3)  +  a 


BY    RATIONALIZATION. 

Since  rational  algebraic  polynomials  and  rational  fractions 
can  always  be  integrated,  an  irrational  differential  may  be  inte- 
grated if  it  can  be  rationalized.  The  rationalization  is  effected 
by  substituting  for  the  variable  of  the  given  differential  a  new 
variable  of  which  it  is  a  function.  Of  these  substitutions  the 
following  are  the  most  important : 

143.  When  the  only  function  of  x  affected  with  fractional 

exponents  is  a  linear  one,  in  which  case  it  will  be  either  of  the 

p  p 

form  X*  or  {ax  +  6)',  assume  x  =  z"  or  ax  +  b  =  z",  n  being  the 
least  common  multiple  of  tlie  denominators  of  the  fractional 
exponents.  For,  if  x=z"  or  ax-}-b  =  z",  the  values  of  x,  ax-\-b, 
dx,  and  the  surds  of  the  given  differential  will  be  rational 
functions  of  z. 


Examples.     1.     | dx. 

x^- 


r 


Here  n  =  12,  and  x  =  z'^. 

Hence  x^  =  z^,  .f'  =  z;  .c-  =  z'',  dx  =  12z^hlz ; 


REDUCTION   BY   NATIONALIZATION.  195 

. ..    r^^--^'  dx  =  f^^  12  z'klz  =  12  Hz'^'  -z')dz 

3.    r    ^^'      =  2  .r  ■  -f-  3  x^  +  <>  x'^  +  6  log (x^  -  1 )  +  C. 

'^  .T-  —  X^ 

^    rcj-xydx _  r(2-x)^-dx 

J        3-x      ~Jl+{2-x)' 
Assume  2  —  x  =  z-. 
Then,       (2  —  x)^  =  z,  dx  =  —  2zdz, 

and  r(2.n^^=  rZL2^  =  _2  ffl L_U 

J        3~x  J     1+z-  J  \        l-\-z-J 

=  -2(2  +  fot  'z)+C 

=  -  2(cot-  ^/iT^  4-  ViT^-)  +  C. 

J  {2r-yY- 


/ 


dx 


=  3[(a;-f- l)i  +  2(;c+l)*+21og((a;4-l)*-l)]  +  C. 
7.  ^:>?{l  +  x)^dx  =  2(\■Vx)\{\^-xy-■\{\^-x)^\-\^C. 

144.   When  the  only  surd  of  the  given  differential  is  of  the 
form  Va  +  &x  ±  ic^,  rationalization  is  effected  as  follows : 


I.    When  the  sign  of  y?  is  jjositive,  place  Va  +  hx  +  x^  =  z  —  x. 

Then        a-\-hx  =  z-  —  2zx; 

,  z^—a      ,        2(z-  +  bz  -i-a)dz 

whence         j:  = ,    dx  =  —^ ' —-^ — , 

b  +  2z  {b-{-2zy      ' 


196  THE  INTEGRAL  CALCULUS. 

and  Va  +  hx  +  x"  =  z-x  =  ^''  +  ^^  +  ^. 

b+2z 

The  given  differential  will  then  be  a  rational  function  of  z, 
since  x,  dx,  and  Va  +  bx-\-oif  are  rational  functions  of  z. 

II.    Wkeyi  the  sign  ofxr  is  negative,  place 


Va  ■i-bx  —  2if=  ■y/{x  —  a)  {ft  —  x)  =  (x  —  a)z, 
in  which  a  and  /8  are  the  roots  of  a^  —  bx  —  a  =  0. 
Then        ^  —  x  =  (x  —  a)z-; 

Avhence        x=  ^2^1^   ^^^  =  — T^TXTvr-' 

and  Va  +  to  -  ar'  =  (a;  -  a)z  =  ^^  ~  ")^. 

Z-  +  1 

The  given  differential  will  then  be  a  rational  function  of  z, 
since  x,  dx,  and  ■\fa  +  bx^^^  are  rational  functions  of  z. 

dx 


Examples. 


/ 


Vl  +  a;  +  ^2 


Assume   Vl  +a;  +  a^  =  2!  —  a;. 

z'-\ 


Then 


1+22 


whence        d.r  =  ^(^l+^+H^. 

0^  +  2zy 


Hence 


Vl  +  a;  +  a^  =  2  -  a;  =  ?-±^-:t_l. 
l  +  2z 

C         '^^  f-^^  =  log(l4-22)  +  C 


=  log  (1  +  2  a;  -I-  2  Vl  +  a;  +  a«)  -H  C. 
2.  J*^-^— ^  =  log {2Vx'-x-l  +  2x -  1)  +C. 

»/  ar  ',._i-A/9^a.ri'^ 


:+V2x-{-x' 


REDUCTION    BY   RATIONALIZATION.  197 

4.  f ^ =  f-^  =  iog(r+x)+a 

Or,  by  the  above  method, 

f  ^^       -  =  log(l  +  a;  +  Vl+2a;  +  x-2)  _^ ^« 

=  log2(l+a:)+C'. 
Prove  that  C'=C-log2. 

5.  r-,^  =  iog(i  +  x  +  V^T^)  +  a 


V2  4-  a;  -  a^ 
The  roots  of  x*^  —  a;  —  2  =  0  are  2  and  —  1. 
Hence     x" -x-2  =  {x -2){x  +  l), 
and  V2  +  X  -  a.-2  =  V(x-  + 1)  (2  -  a;)  =  {x  +  l)z. 

Squaring,  we  find  x  —  "^^ — -  ;  whence 

dx=  ,  0  ,  ...,,    ^2-irX-xr  =  {x  +  l)z=^—-' 
{z-->riy  z^-\-l 

Hence       f— =^^==  f- -^  =  2  cot  ^z  +  C 
J  V2  +  a;-a;2     -^       ^  +^ 

=  2cot-\/2zi^  +  a 
^ix  +  l 


r.  f      ^^       =2cot-'J^ 

»/   -x/2  -  rr  -  ar^  \a; 


—  a 


V2  -  a;  -  ar'  \a;  +  2 

f  ^^  =2cot-\/I 


+a 


8.  i        ""^       ^2cot-^^:^  +  a 

V4a;-3-ar'  \a;-3 

9.  f       ^^-       =2cot-\/^^-^-^+a 

-^  V2-2a;-ar^  VS  +  l+a; 


198  THE  INTEGRAL  CALCULUS. 

145.   Binomial  Differentials.     Every  binomial  differential 
may  be  reduced  to  the  form 

of  {a  +  bx"ydx, 

in  which  p  may  be  any  fraction,  but  m  and  n  are  integral  and 
n  positive. 

For,  let  x^{ax^-\-bxfydx  be  the  binomial,  and  let  k  <  t. 

Then 

a;*(a.r*  +  bx^ydx  =  x^x^'fa  +  6-Yda;  =  x-''+^*(a  +  bx^'^ydx; 

in  which  t  —  k  may  be  fractional,  but  is  positive,  and  h  -\-pk  is 
fractional  or  integral,  positive  or  negative.  That  is,  the  bino- 
mial is  of  the  form 

X  ''(a  +  bx  fydx. 

Put  x  =  z^,  and  this  becomes 

z^'\a  +  bz+<"ydfz^^-'dz  =  dfz^'^+'V~\a  +  bz^^'ydz, 

in  v'hich  ±  cf-\-  df—  1  and  de  are  integral,  and  the  latter  posi- 
tive. 

Hence  writing  m  for  the  former  and  n  for  the  latter,  we 
have  dfz'^{a  +  bz"ydz,  which  is  of  the  required  form.     As  p 

may  be  fractional,  represent  it  by  -, 

s 
We  are  now  to  show  that  a  binomial  differential  of  the  form 

r 

af  (a4-6a;'')'dic,  in  which  m  and  n  are  integral  and  n  positive, 
may  be  rationalized,  and  therefore  integrated : 

I.  When  is   a   whole   number  or  zero,  by  assuming 

a  +  bx"  =  z". 

II.  Wheyi 1-  -  is  a  whole  number  or  zero,  by  assuming 

a  -f  bx"  =  z'x". 

To  prove  that  the  rationalization  is  effected  when  the  above 
conditions  are  satisfied : 


REDUCTION   BY   RATIONALIZATION.  199 

I.  Let  a  +  bx"  =  z\ 

Then  .  =  (£^)i   ..=  (--^)",  <^=i(-_^y-.-... 

and         (tt +  te")''  =  2;''. 
Hence 

\     0     J  nb\     h     J 

=ni>'    \-ir)     ^"^ 

which  is  rational  when  — '^^^-  is  a  whole  number  or  zero. 

n 

II.  Let  a  +  bx"  =  z'x". 
Then        x  =  f )",  .f"'  = 


dx  =  -^(^LS;''-^^dz, 
n\z'-b)      {z'-by    ' 


and  (a  4-  bx")'  =  z'x'  =  z''( — - — V 

Hence      a;'"  ( a  -f-  bx" )  'd.t^ 


«   \i:   J_a_. >^  .  ^/^_^\^-' _^±_ az 


f  —  bj        \z'  —  bj      n\z'—bj      {z'  —  b)- 

=  --a"     'z'^M—±—]«     '     dz, 
n  yz'  —  bj 

which  is  rational  when  — — — h  -  is  a  whole  number  or  zero. 

r  .  n  s  r 

When  -  is  a  positive  integer,  the  factor  {a  +  bx")'  may  be  ex- 
panded and  integrated  directly. 

Examples.     1.    C — ^^^      =  Cx'(a-\-baf)-idx. 
^'  {a-\-b^)^    ^ 

Here        *A±i  =  2. 


200  THE  INTEGRAL  CALCULUS. 

Assume  therefore 

a  -|-  6ar  =  2- ;    whence   (a  +  bx-)  -  =  z^, 


Hence 


r     x'dx      _  r/z^  -  a 
^  {a  +  bx'y^    ^\    ^    . 


b^{z^-ay- 

zdz 


b^z'-ay- 

^1  2a  +  bx^   ^ 

^Wa  +  b^ 


3.  Cx(l  +  xf^dx=  ^(1  +  x)^{rix-  2)  +  C. 

4.  r — ^L^!:L_  =  ^^(a  4.  bx:'y^dx. 
^  {a  +  bx-y~     ^ 

Here        !'i±i +!=_!. 
n  s 

Assume  therefore 

a  -\-  6ar'  =  z'-'or ;    whence  a^ 


Hence 


(a  +  bx')-'  =  ^(-^^—\\    dx=- 

/x-^rfx-  _  r    a  (i-zdz  (z^  —  6X1 1 

(tt  +  6.r2)'~     Jz^-?,     (22_5)|  V    a    y   ar' 
rdz        a     ,  ^i      a         .t''  ,  ^ 


{a-i-bxy- 


5.  r — ^^^ —    liog^ — +e. 


/ 


REDUCTION   BY   PARTS,  201 

dx  1     a  +  2  bar 


7f(a  +  bx")  ^  «"  x{a  +  bar)  ^' 

7.  f ^^=^  =  -l±Mvr^=i>^'+a 

8.    Cx\l  +  2  a^)  ^dr  =  ( 1  +  2  af')  ^  '^^^-  +  C. 

BY    PARTS. 

146.  Let  7c  and  ^•  be  any  functions  of  x.     Then 

d(^tiv)  =  udv  -\-  vdu. 
Transposing  and  integrating, 

I  iidc  =  nv  —  I  vdu. 

This  formula  is  known  as  the  formula  for  integration  by 
parts.  It  evidently  makes  the  integration  of  udv  to  depend 
upon  that  of  vdu.  To  apply  it,  the  given  differential  must  be 
resolved  into  factors  m  and  dv  such  that  dv  and  vdu  shall  be 
integrable.  The  following  are  the  most  important  applications 
of  this  formula. 

147.  Binomial  differentials.     Formulae  of  reduction. 

It  has  been  shown  that  every  binomial  differential  may  be 
reduced  to  the  form  x"'(a  +  bx^ydx,  in  which  p  is  any  fraction, 
but  m  and  n  are  integral  and  n  positive. 

I.   Let  u  =  x"'~"+\  dv  =  {a -\- bx")"x''-^dx. 

Then    du  =  (m  —  n  -\-  l)x"*~"dx,     v  =  ^^      trr  * 

nb{p  +  l) 


Substituting  these  in    |  udv  =  uv  —  i  vdu, 

I  af(a  +  bx")Pdx  = -^^ — ' — —^ — 

- 'm'-n-\-\  r  „_„ .        hx'^y^'dx. 


202  THE   INTEGRAL   CALCULUS. 

But 

I  ^-»(^ci  +  bx")P^kl.r  =   i  .T'"  "(a  +  bx"y{a  -\-  hx'')dx 

Hence 

or,  solving  for    j  af  (a  +  bx"ydx, 

Cx'^ia-^bx^ydx 

x'"~"+\a  4-  bx"y^^  —  a{m  —  n-^1)  Cx"'"{a  +  bx^ydx 

~ TT — ; r~r\ '  ('^^ 

b{np-\-  m  4-1) 

a  formula  which  makes  the  integration  of  the  given  binomial 
to  depend  upon  that  of  another  in  which  the  exponent  of  the 
variable  without  the  parenthesis  is  diminished  by  that  of  the 
variable  within. 

Illustratiox.      I —  =  i3if(l—x^)~^dx.      We  apply 

(A)  to  this  differential  because  its  integration  would  thereby 
be  made  to  depend  upon  that  of  x{l  —  x^y-dx,  which  comes 
under  Form  1.  Substituting  therefore  in  (-4)  m  =  3,  n  —  2, 
p  =  —  ^,a  =  l,  6  =  —  1,  Ave  have 

^                             x'(l-a^)^-2Cx(l-a^)~^dx 
J  x'{l  -  x'yhlx  = 4, : — • 

=  _  ^ic2(l  _  a;2)  2  _|_  I  r^il  -  x'ykx 


REDUCTION    BY   PARTS.  203 

If  ?»p  -j-  m  4-1  =  0,  the  formula  fails ;  hut  in  this  case 

m  +  1 
n 

and  the  differential  may  be  rationalized  and  integrate4  by 
Art.  145. 

II.     (  af  (a  +  hx^ydx 

=  Cx"'{a  +  hx")"~\a  +  I)af)dx 

=  a  Cx"'{a  +  bx")"  \1x  +  b  Cx"^"{a  +  bx''y-'^dx.        (1) 
Applying  (A)  to  the  last  integral  of  (1),  we  obtain 

Cxr^"(a  +  bx"y~^dx 

3r-^\a  +  ba^'Y  —  a{m  -f  1)  j  a-"'(a  +  bx^y^Hx 
b{np  -i-m  +1) 
Avhich,  substituted  in  (1),  gives 

I  (ir{a  +  bx!^ydx 

yf'+^a  -f  6a?")''  -f-  anp  Cx'^ia  +  bx^y-Hx 


(B) 


vp  +  m  -\- 1 


a  formula  which  makes  the  integration  of  the  given  binomial 
to  depend  upon  that  of  another  in  which  the  exponent  of  the 
parenthesis  is  diminished  by  1. 

Illustration.      |  (a^-far)^dx.     The  application  of  (B)  to 
this  differential  makes  the  integration  depend  upon  that  of 
^  ^     -,  which  can  be  rationalized  and  integrated  by  Art.  144. 


Va"  -I-  x-2 

Substituting,  therefore,  in   (B)   m  =  0,  n  =  2.  p  =  ^,  a  =  a*, 
6  =  1,  we  have 


204  THE   INTEGRAL   CALCITLUS. 

dx 


fia'  +  x'ydx 


•^  Va^  +  x" 


Writing    Va^  -(-  ar  =  2  —  a-,  we  find 

r^x__  ^  rdz^iQgz+C  =  log  (.T  +  V^N=^)  -f  C. 

Hence 

C(a^  +  ar') ^. dic  =  ^ x(a-  +  ar')  ^  +  ^  log  (ic  +  V^^H^)  +  C. 

If  rjj9  +  m  4- 1  =  0,  the  formula  fails,  but  Art.  145  applies  as 
before. 

III.    In  {A)  let  m  =  m-\-n.     Then 

Cxr+"{a  +  bx''ydx 

x^+^a  +  bx''y+^—  a{m  + 1)  j  a;'"(a  +  6a;")''da; 

6(np  +  m  +  n  + 1) 
whence 

I  a;"'(a  +  bx^ydx 

a  formula  which  makes  the  integration  of  the  given  binomial 
to  depend  upon  that  of  another  in  which  the  exponent  of  the 
variable  without  the  parenthesis  is  increased  by  that  of  the 
variable  within. 

lLT.USTRATIOJf 


r ^ =  Cx  %x'  -  l)-^rfa;.      By  ap- 

;egration  is  made  to  depend  upon  that  of 
,  which  is  a  known  form.     Hence,  making  m  =  —  3, 


plying  (C)  the  integration  is  made  to  depend  upon  that  of 
dx 

xy/oi?  —  1 

n  =  2,  /)  =  —  ^,  a  =  —  1,  6  =  1,  in  (C),  we  have 


/: 


dx 


REDUCTION   BY   PARTS.  205 


If  m  =  —  1,  the  formula  fails  ;  but  in  this  case  =  0, 

and  Art.  145  applies. 

lY.    In  {B)  let  2^=p-\-l.     Then 
\^{a-\-hx''y^Hx 

— . d , 

np  -\- n  +  m  -\- 1 

whence 

fee"  (a  +  bx"ydx 

—x"'+\a-\-bx"y+'^+  (wp+w-l-m+l)  Cor(a-{-bx^y+^dx 


an{p-{-l) 


AD) 


a  formula  which  makes  the  integration  of  the  given  binomial 
to  depend  upon  that  of  another  in  which  the  exponent  of  the 
parenthesis  is  increased  by  1. 

r      dx  C 

Illustration.     I— ;r5=  I  {\-\-^y^dx.     By  applying 

./  (1  +  xry     J 

(D)  twice,  we  see  the  integration  will  be  made  to  depend  upon 

that  of  ?  which  is  a  known  form.    Hence,  making  wi  =  0, 

1  +aH 


H  =  1,  2>  =  —  3,  a  =  6  =  1,  in  (D),  we  have 
""4(1+0^)^  '  4 


^^^        -  x{i-\-x^y'-  3  f(i-\-x'y'-dx 

(1+^'"  =^4 


206  THE  INTEGRAL  CALCULUS. 

Applying  {D)  to  the  last  integral,  we  have  m  =  0    «  =  <> 
i;  =  -  2,  a  =  6  =  1,  and  '  ' 


f(^+^rM.^Z^^^^lfnzSj^±^ 


Hx 


^'    ^+htiin  \v.+a 


2(1  +  ^) 

Hence 

Examples. 

'^  {of  —  Qi?y-  ^  2,  a 

yd    -\-  OCT  J  -        -* 

^-  J  (a-'-  ar')  kx  =  ^(a^  -  o-^^  +  ^^'  (a^  ^-  o;^)* 
**  8 

,3  a''.     ,  a;      ^ 

+  — -sin->-+C. 

Appl'y  (5)  twice. 


EEDUCTION   BY   PARTS.  207 

8.  Cx'{l-af)klx=  -^(^  ~  ^)  '+^(^  -  ^^y+^sin-'x  +  C. 
J         ■  4  8 

Apply  (yA)  and  (B)  in  succession. 

o      C       3?dx  C  3  N-'^ 

9.  I  —  =  I  .T-  (2 a  —  .t)   -  ax 

*^  V2  ox  —  ar'     '^ 

= ^^ (2 aa;  —  ar)  -  H rers^  -  +  C. 

2      ^  ^2  a 

Apply  iyA)  twice. 

10.   f '^ =  -^^'-^  +  c. 

Apply  (C). 


11.  f ^ =  -^^^^^^-4-^ log  -fC. 

-'ar'(l-.r2)^  S-r^  Vl  -  a:'^  + 1 

12.  f ^ = ? +J^tan-'^  +  C. 

J(o;  +  ar^)2      2aXa^  +  iK')      2  a''  a 

13.  Show  that  (^)  will  reduce  the  following  to  known  forms  : 
,  if  m  is  even  and  positive ;  also  if  m  is  odd  and 


Va^  —  a^ 
positive. 

x'^dx 


,  if  m  is  even  and  positive. 

/~~i~t — « 
Va'^  +  ar 

±? 
^(o?  ±  a^)  »,  if  m  is  odd  and  positive. 

What  if  m  is  odd  and  negative  ? 

14.    r(r2  -^)^dx  =  \x{i~  -  x-') ^  -h  ^ r^ sin^^  ~+C. 

15.  r  y'^    =-^y^+^Ky+^^)v2^^j7^^ 

v'  -s/lry  —  f  ^> 

-l-fr'vers-i^  +  C. 


208  ,  THE  INTEGRAL  CALCULUS. 

148.   Logarithmic  differentials  of  the  form  (jc*** (log  x)ndx 

may  be  integrated  by  parts  when  n  is  a  positive  integer,  by 
placing  af*dx  =  dv,  (log  xY=u,  in  the  formula 


I  udv  =  vv  —  I  vdu  ; 


every  application  of  the  formula  reducing  the  exponent  of  the 
logarithm  by  unity  and  thus  finally  making  the  integration 


depend  upon  |  x'^dx. 

Examples.     1.    \  x- {\o^  xydx. 

Let  a?dx  =  dv,  (loga;)-=  n.    Then  v  =  -,  du  =  2  log  x — ,  and 

3  X 

Jtidv  =  (log  xY I  ar^log  xdx. 

Placing  ^dx  =  dv,  u  =  log  x,  whence  v  =  —,  du  =  — , 

3  X 

J  udv  =  -  log  a.-  —  -  I  x'dx  =  ~  \ogx  —  ^  +  C. 
Hence  ("^^(log  x)Hx  =  -  [ (log  x)--  f  log  a;  -(-  f  ]  +  C. 

2.  I  log  xdx  =  X  (log  x  —  \)-\-C. 

3.  Jar'(loga;)^rf.r  =  ^'[(loga;)^'-iloga;  +  i]  +  C. 

149.  Exponential  differentials  of  the  form  x^^tF^dx  may 

be  integrated  by  parts  when  n  is  a  positive  integer,  by  placing 

a;"=  XL,  e'"dx  =  dv,  in  I  udv  =  tiv  —  |  vdu,  every  application  of 
the  formula  reducing  the  exponent  of  x"  by  unity,  and  thus 
finally  making  the  integration  depend  upon    |  e'"dx. 


REDUCTION    BY   PARTS.  209 

Examples.     1.    |  are^'xdx. 

Let  e'"dx  =  dv,  x'=u\  then  v  =  — ,  dn  =  2  xdx,  and 

a 

/udv  = I  e'^xdx. 

Placing  e'"dx  =  dv,  x  =  m,  whence  v  =  — ,  die  =  dx, 

a 

erxdx  =^^  -  -  I  e-fte  =  ^  -  £-  +  C. 
a        aJ  a       or 

Hence   (W^d.c  =  — ^3?^-^  +  ^  V  C. 
J  a  \  a       o?j 

»/  \<i        a^       cr      ay 

150.  Trigonometric  Differentials.  By  simple  transforma- 
tions, some  of  which  are  indicated  in  the  following  examples, 
these  may  often  be  reduced  to  known  forms.  Otherwise  resort 
must  be  had  to  integration  by  parts. 

I.   sin»»  xdx  and  cos"  xdx. 

(a)  When  n  is  an  odd  integer,  we  may  write 

n-l 

sin^o^a;  =  (1  —  cos^x)  "^  sin  xdx, 
and  cos"  xdx  =  (1  —  sin^  x)  ^  cos  xdx. 

1.  I  sin®  xdx  =  I  (1  —  cos''  x)  sin  xdx  =  —  cos  x-\-^  cos'  x-\-C. 

2.  jcos*a;dx=  |  (1  —  sin^ic)''cosa;diB 

=  sin  a;  —  I  sin®  x+\  sin*  x-\-C. 

3.  I  cos®  xdx  =  sin  x  —  ^  sin®  x  +  C. 
{b)    When  n  =  2,  since 

2  sin'  a;  =  1  —  cos  2  a;,  and  2  cos^  a;  =  1  -j-  cos  2  «, 


xdx 


210  THE  INTEGRAL  CALCULUS. 

4.  i  sill' xdx  =  y{-^  —  ^cos2x)dx  =  -  —  ^sm2x  -\-C. 

5.  I  cos^  xdx  =  f  +  }  sin  2x-\-C. 

(c)   hi  general,  when  n  is  any  integer,  let 
}i  =  sin"  'a;,   dv  =  sin  xdx. 

Substituting  in    |  udv  =  uv  —  |  vdu,  we  have 

I  sin''ccda;  =  —  sin""^;«cos.c  +  (n  —  1)  |  eos^icsin"  -a;da; 

=  — sin""^a;cos  CC  + (n  — 1)  |  (1  —  sin^a;)sin" 

=  —  sin"  ^ccces  x  +  {n  —  1)  j  sin""'^.'Kda;  —  (n  —  1) 

I  sin"  xdx. 

Transposing  the  last  term  to  the  first  member, 

/.   ,,     ,            siu"~^a;cosic  ,  ?i  —  1  /^  .   ,,  ,    , 
sin"  xdx  = 1 I  sin"  ^xdx. 
n                   n    J 

The  integration  is  thus  finally  made  to  depend  upon  I  dx=x 
if  n  is  even,  or  upon   j  sin  xdx  =  —  cos  x  if  n  is  odd. 
In  like  manner, 

/,,     ,        COS""'  X  sin  x  ,  n  —  1  r      ,,  .,     , 
cos"  xdx  = 1 I  cos"  -  xdx, 
n                  n     J 

the  integration  depending  on    |  dx  =  x  if  n  is  even,  or  upon 
I  cos  xdx  =  sin  x  if  n  is  odd. 


r  ■   A     J            sin^  X  cos  X      'S  r  ■   o     , 
).    I  siir  xdx  = +  -  I  siir  xdx 

—  f  sin  X  cos  X  -\-^x-\-C. 


sin'^xcoscc      Sr     sin  ic  cos  a? 

+  7  t: r; 


4  41  2 

sin''  X  cos  X 


REDUCTION    BY    PARTS.  211 

7.  Ccos*xdx  =  "°^'  a;  «in  X  _^  ^  ^-^^  xgosx+^x  +  C. 

8.  fcos"  xdx  =  ^"^^  ^'.^^"  ^'  +  ^  cos'^  X  sin  a; 

»^  "4-  i|sin  a? cos  x  +  |f  ic  +  C. 

II.    -.—n —  and 


sin**  a-'  COS"  a? 

(«)    ir^en  ?i  ts  an  even  integer,  we  may  write 


dx 

sin"  x 


n-2 


=  cosec"^  ^  aj  cosec-  xdx  =  ( 1 + cot^  a;)  2  cosec'  xdx, 


and  =(l  +  tan-a;)  2  sec^icdr. 

cos"  .T 


9.    (*  ^•^'    =  f  cosec*  ;k  cosec^  xdx  =  i  ( 1  +  cot^  x)  ^  cosec''  xdx 
J  sin"  a;     J  J 

=  —  cot  a;  —  I  cot''  x  —  l  cot*x  +  C. 

10.  r_^  =  tan  a;  +  ^  tan^  a;  +  C. 
J  cos*  a; 

11.  r_^  =  tan  a;  + 1  tan''  x  +  i  tan^  x-\-C. 
J  cos^x 

I* 

(6)    When  n  is  1,  we  have 


dx 


2cos2!^          Asec=^-da; 

12.  r_^=  f__j^i!-_=  f ^=  r.    ' 


2  sin -cos-     *'      sin-       *^        tan^ 
2       2.  2  2 

X 

cos- 
2 

=  log  tan  1  +  0. 


13.    C^^C       f-'         =  -  log  tan  (l-?^  +  (7,  by  Ex.  12. 


212  THE   INTEGRAL   CALCULUS, 

(c)   In  general,  when  n  is  any  integer, 

/dx    _  /^cos^ X  4-  sin^ x  ,  ,_  f         cos xdx       C 
sin"  re     J  sin"x  J  sin"ic      J  i 

Let      u  =  cos  X,   dv 


dx 


siii''~''a; 
cos  xdx 


sm"  X 


Substituting  in   i  udv  =  uv  —  |  vdu,  we  have 


fcos  xdx                  cos  x                 1      C    da. 
cos  .T — ; = : ■ I  
sin"a;            (n  — l)sin"  'a;      n  —  1»/  sin"" 


dx 
(n  — l)sin"  'a;      n  —  1 J  sin"~''a; 


n  —  '2r    dx 


Hence   CJ^  = 521£ +  !iZl^  ( 

J  sin";r;  («  —  1)  sin"~^a;      n  —  1»/  sin"~^a; 

The  integration  is  thus  finally  made  to  depend  upon 

/.   ,    =  I  cosec^a^ic  =  —  cot  x, 
sin^'ic     J 

if  n  is  even,  or  upon 

if  n  is  odd. 

In  like  manner, 


/dx    _  sing; ^  »  —  2  T 

cos" a;      (n  —  l)cos'*  ^'c      n  —  iJ  c 


dx 


(n  —  l)cos'*  ^'c      n  —  1.7  cos"~*ic 
the  integration  depending  upon 

— —  =  I  sed^xdx  =  tan  x, 
,    cos-'x     J 

if  n  is  even,  or  iipon 

C-^  =-logtanf^-?VEx.  13), 
J  cos  if  \^4      2/ 

if  n  is  odd. 


J  sin*  a;          4sin*a;      iJ  i 


REDUCTION   BY   PARTS.  21  o 

dx 


snrx 


cos 
4  sin 


)s  a;        '^/'_  cosx     i^r^^\ 
in^iK      4',^     2sin^a;      2»/  sin .»/ 

3  cos  a;   ,  3 1      ,      a;  ,  ^ 
4- -log  tan- +  C. 


4  sin"*  X      8  sin^  x      S  2 

J  cos-^a;     2cos2.x      2    ^        V"^      27 

TTT     BinV^xdx        1  cos"icrfa!; 

ill.    and 

cos"»ic  sin'"^ic 

(a)  When  n  =  1,  we  have  directly,  by  Form  1, 

16.  - — ^- dx  =  —  I  (cos  x)~'(—  sin .rr/a*)  = ^- C. 

cos'  X  J  6  cos^  a; 

J  sm*  a;  3  sin^  x 

(b)  When  n  —  m  ?"s  negatice  and   even,  Form   1   applies   if 
we  write 

sin"  xdx 


cos   x 


=  tan"  X  sec"'  "  xdx, 


or,.i  cos"a;da;  ,„  __„    , 

ana  —, =  cot"  a;  cosec"  '^xdx. 

sin*"  a; 

IS.    I  - — ^dx  =  I  tan^a;  sec^a;dx  =  A  tan*x-f-C. 
J  cos'  X  J 

19,  I  dx=.  j  cof  .r  cosec' aida; 

J  sm'a;  J 

=  I  cot''a;(l +  cot*a;)^cosec^a;daj 
=  — ^cot^a;  —  ^cot^a;  —  ^cot'a;  -|-C. 

20.  f?lB!^"=itan'^x  +  a 
J     cos*  a; 


214  THE   INTEGRAL   CALCULUS. 

21.  C^-^^  =  -^cot^x  +  a 

22.  C^^  dx  =  I  tair^  x -\- 1  tan'  a;  +  i  tau»  x  +  C. 

(c)  TFZieii  w  —  m  is  negative  and  odd,  ifn  is  odd,  we  have 

dx  =  tan"  X  sec"'  "ccdx 

cos^a; 

=  (sec^x  —  1)  ^  sec"  ""'.r  tan  x  sec  xcte, 

to  which  Form  1  applies,  and  ^^^  ^^^  may  be  treated  in  a 
similar  manner. 

23.  I  — ~--dx=  I  tan^iKsec^ardic 
J  cos'" a;         J 

=  I  (sec-  x  —  iy  sec^  x tan  x  sec  xda; 

=  I  (sec^ic  — 2sec''.x'  + sec^ic)tanxseca;dx 
=  ^  sec^  x  —  l  sec^  .r  +  i  sec^  x-{-C. 

24.  f?H^==  isec*cc-4sec-'.<;+C. 
J    cos^a;        ^  ^ 

25.  I  -r— r-t?iK  =  — 4cosec^a;  +  coseca;  +  C. 
J  sin^a;  ^ 

(d)  IF^en  n  —  m  is  jiositive,  resort  must  be  had  to  integra- 
tion by  parts.     When,  however,  m  —  n  =  1,  and  n  is  odd, 

nn     /'sin^  xdx       /* .  -,  2   \  si"  ^  ^  .  ,  n 

^D.    I — =  I  (1— cos'^a;) — —  da;  =  sec  a;  +  cos  a; +  C. 

J     cos'^x       J  cos^'a; 

o7     rcos^a;da;  .         ,  ^ 

Li.    I  — r—^ —  =  —  cosec  X  —  sm  a;  +  (7. 
J     sin-  X 

OQ     Tsin^a^da;  1  1       ,      .3      ,  ,  ^ 

28.    I —  = —  -\ -I-  cos  a;  +  C. 

J    cos"  a;       5cos*x     cos-*  a;     cos  a; 


REDUCTION   BY   PARTS.  215 

IV.   tB.n."*^xdx  and  cot"^  ocdjc.      These  forms  can  be  inte- 
grated directly,  when  m  is  integral  and  positive,  by  placing 

tan"*  icda;  =  (sec^x*  —  l)tan'"~^icdx, 

and  cot"*  axlx  =  (cosec^  x  —  1)  cot""^^  xdx. 

20.    I  tan^ xdx  =  |  (sec- x  —  l)dx  =  tan  x  —  x-{-C. 

30.  Ctai,n^xdx=  j  (sec^a;  — l)tani»daj  =  ^tan''a;—  it&nxdx 

=  A  tan-  X—  (  ^^2_?  dx  =  \  tan^  a;  4- log  cos  x+C. 
J  cosx 

31 .  I  tan^  xdx  =  |  ( sec-  .x  —  1 )  tan^  xdx  =  \  tan''  ic  —  I  tan^  icrfjj 

=  ^ tannic  -  tan  a;  +  x  +  C  (Ex.  29). 

32.  I  tan*  ardx  =  \  tan*  a;  —  ^  tan-  x  —  log  cos  a;  +  C. 

33.  I  cot^a^a;  =  —  cot  a;  —  a;  -f-  C'. 

34.  I  cot^  xdx  =  —  ^  cot^  X  —  log  sin  x+C. 

35.  I  cot"  xdx  =  —  i  cot*  x  +  ^  cot^  a;  +  log  sin  x  +  C. 

36.  I  (tan^ »  + tan* a;)da;=  |  tan^ a; sec^ xdx  =  \ tan^ a;  +  C. 

37.  j  (tan^  x  +  tan*  x)dx=  |  tan^  x  sec-  xc?x  =  \  tan^  x  +  C. 
And,  in  like  manner,  (tan"x  +  tan"'x)dx  when  ?i  —  m  =  2. 

V.  x"siii(aa7)<fic,  and  a;"  cos  (aa?)  da?. 

Let  w  =  x",  dy  =  sin  (ax)dx. 
Substituting  in  I  iidv  =  uv  —  \  vdu, 

fx^sin  (ax)dx  =  _?!£2l(^  +  !-^  fcos  (ax)x"-idx, 


^^^  THE   INTEGRAL   CALCULUS, 

the  integration  finally  dei^ending  upon 

J  cos  {ax)dx  or  jsin  {ax)dx. 

38.  J  x-^cos  xtJx  =  ;r'sin  x  —  3  Cx-ain  xdx 

=  x-^  sin  a;  -  3  ( -  a-2  cos  a;  -  2  J-  x  cos  xdx) 
=  ay  sin  a;  +  3  x-  cos  a;  —  G(x  sin  .c  -  Tsin  xdx) 
=  ^'shix-\-3x^cosx-6xsmx~6co8x+a 

39.  J  a^  sin  a;da;  =  -  ar' cos  a;  +  2  x  sin  a;  +  2  cos  a^  +  (7. 

4( ).    fx  sin  (ma;) da-  =  -  "'^li'"^  C^"-^)  _,_  ^i"  (m-^0    ,   ^ 

VI.  €«*  sin"  xdx,  and  e"-*  cos"  xdx. 

Let  M  =  sin".r,  dv  =  e<"da-. 
Substituting  in  Cudv  =  uv  -  fvdu, 

In  the  last  integral  let  u  =  sin'-^.a;  cos  x,  dv  =  e^^d.x.     Then 
du  =  (n  - 1)  sin'-2.c  cos'xdx  -  sm^xdx 

=  (n  -  1)  sin'-2_^.(i  _  sin2^)rfa;  -  sin"a;c?a;. 
=  (h  —  1)  sin"-2a;da;  —  n  sin'' xdx, 


and  the  formula  Cndv  =  uv  -  fvdu  gives 
J  e^sin"-^  a;  cos  a;da; 

- '-^ J'e'-sin"  ^a-f^,^^ n  p<«sin«,p^^_ 


_  sm"'^a;cos  xe"^     n  —  1 
a 


REDUCTION    BY   PARTS.  217 

Substituting  in  (1)  and  solving  for  j  e"su\"xdx, 

=  ^r^iS!:!^  (a  sin  X  -  n  cos  x)  +  ''^['-^  fe-sin"  ^xd..    (2) 

By  a  repetition  of  this  process  the  integration  is  made 
to  depend  upon  the  known  form  i  e'^'dx,  or  \ipon  ie"''smxdx, 
which  bv  (2)  is  — —  (a  sin  a;  — cos  a;),  n  being  1.  From  the 
form  e"-'cos".Tda;  Ave  liave  in  like  manner. 

I  e"*cos"icda; 

e^'sin  a;d;c  =  — — -   (a  sin  x  —  cos  a;)  +  O. 
a^+l 

42.  f  e"  cos-  xdx  =  ^!!^2i^  (a  cos  or + 2  sin  x)  -\ ^-^^  +  C. 

J  a'^+4  o(a-+4) 

43.  j  e'  sin^ xdx  =  —  (sin''  a;  -f  P>  cos'' a;  +  3  sin  a;  —  G  cos  a; )  -}-  C. 

151.  Circular  differentials  of  the  forms  /  (a?)  sin^  icrio?, 
/(£r)  cos  ^  xrfic,  etc.,  (7i  iohichf{x)  is  an  algebraic  function. 

Assuming  dv=f{x)dx,  the  formula  for  integration  by  parts 
will  make  the  integration  depend  upon  an  algebraic  form. 

Examples.     1.    I  sin  ^Trfa*. 

If  =  sin~'  X,  dv  =  dx,  da  =  —    '  v  =  x.     Then 


/sin^^  xdx  =  a;  siu"^  -^  ~  I  — ^  =  ^'  ^^^'^  ^'  +  (1  —  •'»^) "  + 

2.    ftan-i  a;da;  =  x  tau"^  a?  —  |  log  ( 1  +  a;-)  +  C. 


218  THE   INTEGRAL   CALCULUS. 


0.  j  ar'cos-^ xdx=  ~  cos  ^ x  -  ^^ ^     ^^  (a'-+  2)  +  0. 

4.    I  xQOii-^xdx=^x^cos~^x  —  ^x(l  —  af')-  +|^sin^*a;  +  C. 

BY   SUBSTITUTION. 

This  method  has  been  already  employed  in  the  rationaliza- 
tion of  irrational  differentials  (Arts.  143-4),  and  consists  in 
substituting  for  the  variable  of  the  given  differential  a  new- 
variable  of  which  it  is  a  function. 

152.  Trigonometric  functions  of  the  form  sin"  x  cos***xdx. 

1.  Let  sin  a;  =  2.     Then 

sin" X  =  z",  cos" x  =  (l—  z') '',   dx  =  {l—  z') ~^dz. 

Hence  I  sin"a;cos'"a;da' =  |  z"(l  —  z'-)  -  dz, 

or  in  like  manner,  writing  cos  x  =  z, 

I  sin" x cos"'xdx  =  I  —z"'(l  —  z-)  -  dz. 

The  given  differential  may  then  be  integrated  whenever  the 
above  binomials  can  be  integrated. 

Examples.     1.    \sm*xdx.     s,mx  =  z,  dx=:-^~  =  — — — . 
»/  cos  X      -y/i  _  ^ 

fsin* xdx  =  C—^^  =  _  ^  (^2  +  3 )  Vr^^  +  3  sin-i ^^c 
J  J  Vl  -  ^2         4  -  * 

^  (Ex.  2,  Art.  147) 

=  _  c^  (si,^3^  _,_  3  sin  x)  +  f  .^•  +  C. 

2.  rsin^.d.=  r-^=-fi^+ii>8yr^+c 

(Ex.  3,  Art.  147.) 
=  -  ^'  (sin* a;  +  A  sin^a-  +  f)  +  C. 


REDUCTION    BY   SUBSTITUTION.  219 

3.  i  sin* X  cos^  xdx  =  i  z^(l  —z-)-dz  Avhen  sin  a;  =  2. 

fz\l  -  z')^dz  =  -  ^(^ -''')'  +  g(^  -  ^')    + 1  sin-»z  +  a 

(Ex.  8,  Art.  147.) 

-rx             r  •   9         i   J            sinit'cos^a?  ,  sin  a;  cos  a;  ,   ,      ,  ^ 
Hence     I  sin-'a;  cos^a;aic  = 1 f-  ^  .r  +  C. 

II.  When  either  m  or  n  is  odd,  we  may  integrate  directly  by 
treating  the  factor  whose  exponent  is  odd  as  in  Art.  150, 1,,  (a). 

4.  I  sin''.rcos^a;da;=  1  (1  —  cos^x)cos^a;sina;fte 

=  —  ^  COS'^  X-\-\  COS''  X  +  C. 

5.  I  COS"'  X  sin''  xdx  =  ^  sin''  ^  —  ^  sin^  x  +  i  sin^  a'  +  C. 

6.  I  cos^  X  sin^ xdx  =  —  i  cos'  x  -\-  ^  cos"  a;  +  C 

7.  I  sin  a;  cos^  xdia;  =  —  ^  cos^a;  +  C. 

8.  I  cos  X  sin'ajfia;  =  ^  sin®  a;  +  C, 
form  1  applying  when  n  or  m  is  1. 

9.  r_,_J?!? =  ClA^^  =  log  tan  x+a   (Ex.  12,  Art.  150.) 

J  sin  a;  cos  a;     . '  sin  2  .r 

10.  f-_^^_=r?iB!^±^dx  =  tana.-cota.+a 
J  sin^ajcos^a;     ./     sin^ajcos^a; 

11.  I  sin^a;  cos'^xda;  =  \  sm*x  —  ^  sin®  a;  +  C. 

153.  Many  differentials  may  be  integrated  by  substitution, 
but  no  general  rule  can  be  given,  and  the  method  is  best 
exhibited  by  examples,  of  which  a  few  are  added. 


220  THE  INTEGRAL  CALCULUS. 

1-     I  —r-, ^  =1  I  -^-r, :  when  x'  =  z. 

J  x{ci^  +  ar*)      ,)J  z{a^  +  z) 

By  Art.  139,  Case  1, 


J  z{a' 


dz  1 T  z 


+  2!)      a?       (f-\-z 


Hence  i ''^ — -  =  — -log— -^^ — ^,  +  C. 

J  a;(a'  +  .r')       '.\a^       (r  +  .r* 


2.     r___^^_—  =  _  log  2  +  a;  +  2  Var'  +  a;  +  1  _ 
»^  a;Vl  +x-\-  X-  ^ 

Put  a;  =  - ;  the  differential  may  then  be  integrated  by  Art. 
144,  I.         y 

Put  1  +  .-C  =  2. 

Then  f_i±^e=^rfa.  =  Y^^+2  r^'-2  f^^'Y  (1) 

J  (1-f-a;)-  e\         J    z-         J    z   J    ^  ^ 

Placing   M  =  e*  ?a\(i    dv  =  z  -dz,  and   applying   the   formula 

for  integration  by  parts  to    I  — ,  Ave  have 

J     r 

2-  z      J     z 

Substituting  this  value  in  (1),  we  have  the  above  result. 

4.     r^Vl+loga;  =  i  (1  +  logx) '  +  C.     Let  1  +  loga;  =  z. 


f-      C    x^dx  1         ,        a;  Vl  —x^,^     ^    . 

o,     I  — 3—-—-  =  —  A- cos"^^ X h  C\     Let  x  =  cos  2. 


6.     r        d^         _2&^Q^ft  +  to         a  +  25a;         ^ 
Put  X'  =  -,  whence 


x^{a-\-bxy-  (az  +  by 


REDUCTION   BY   SERIES.  221 

In  the  latter  let  az  +  b  =  y,  and  it  becomes ~  ^^~    ^  dy. 

a^       y- 

7.    x^(a-2c^)^dx.     Put  x^=a-z\ 

BY    SERIES. 

154.  When  the  given  differential  can  be  expanded  into  a 
converging  series,  its  integral  may  be  found  by  integrating 
each  term  of  the  series.  The  integral  thus  obtained  will  be  in 
the  form  of  a  series,  and  therefore  integration  by  series  affords 
a  method  of  developing  a  function  where  the  development  of 
the  derivative  is  known. 

EXAMPLKS. 

1.  Cy/x'^^\ix=  CV'x(l-x^)^dx 

=  C^~x(l  --  -~  - ^'-Adx 
J        ^        2       8      If)      ^ 

=  ^x^  -Ix'^-  ^^  cc y  -^x^  ••'  -{-  C. 

2.  C-^dx=  Cn+x  +  x''-{-^a^  +  ^x*'..)dx 
J  cos  a;         ./ 

=  x-  -f- -  +  -  +  -  +  ■'"-  •••  +  C*. 
2       3       0       10 

See  Ex.  18,  Art.  72. 

^.   Develop  log  (1  +  a;) . 

log  (1  +  X)  =  f-^  =  C(l  +  x)-'dx 
J  1  +  x     J 

=  C(l-x-\-x^-x'---)dx 

2      3       4 
4.    Develop  sin~^a?. 

sin-'a;  =  f  /^^      =  C(l  -j-  ^x^  +  |^'  +  H^'  +  ■■•)dx 
•^  Vl  —  X-     "^ 

=  x  +  \x'  +  ^x^  +  jf^^x'  ...  +  C. 


222  THE  INTEGRAL  CALCULUS. 

Eemark.  The  process  of  integration  is  the  inverse  of  that 
of  differentiation ;  but  it  does  not  follow  that,  because  we  can 
differentiate  every  integral,  we  can  integrate  every  differential. 
Suppose,  for  example,  the  given  function  be  a;" ;  its  differential 
is  nx"-^dx.     Now,  in  order  that  the  differential  of  x"  should 

assume  the  form  -,  we  must  have  n  —  1  =  —  1,  or  «  =  0 ;  in 

X 

which  case  x"  =  1,  which  has   no  differential.      That  is,  the 
algebraic  function  x"  cannot  give  rise  to  a  differential  of  the 

form  -^ ;  nor  can  any  other  known  function  except  log  x.     It 

X 

is  evident,  therefore,  that,  before  the  invention  of  logarithms 
and  the  investigation  of  their  properties,  the  operation  indi- 

cated  by   I  —  would  have  been  impossible.     The  transcenden- 

tal  functions  sin"'.'*-,  tan^^a.',  etc.,  whose  differentials 


dx  .  .  ^1-^ 

^,  etc.,  are  algebraic  functions,  are  further  illustrations  of 

l  +  or 

the  fact  that  the  integration  of  algebraic  differentials  may  in- 
volve transcendental,  or  higher,  fvmctions.  The  integration, 
therefore,  of  such  forms  as  do  not  arise  by  the  differentiation 
of  the  known  functions  cannot  be  effected  until  new  functions 
corresponding  to  these  forms  have  been  invented. 


MlSCELLAXKOUS    EXAMPLES. 


Integrate 


-,    1  — x"  ,  J-  1  +  2.1; cos^a;         , 

1.  dx.  5.  —\ dx. 

1  —  x  cosa;  sina;  +  arcos^x 

,^        xdx  ^    _^dx_ 

Va^-x*  •   (l  —  xy 

•  '7T~, — Ta"  T.  xt&w^xdx. 


{1-^xy 

4.  ^(^±1  dx.  8- 


bdx 


■y^y.  _  1      '  Vc"^  —  a^  —  2  abx  —  b^oi^ 


MISCELLANEOUS   EXAMPLES.  223 

Q •  ■"'J-   ; oTT.  "*^' 

V3-(ix-<^ar  («  +  ^^  +  «^) 


1  +  ^  f^f,      sin  Ticcdaj 


11.  '^^ 


a*  —  x* 


X*  —  a' 


7ji  —  COS  r?a; 


fiinxcos'x  sec^xdx 

26.  7 

■>      1  7rt  —  n  tan  .^• 
12.       ^  ~^      dx. 

x^^.r'  +  l  nx"hlx 


27. 


da;  Va'"-a;"-"' 


on      cos  xda; 


a-daj  «^  +  sin-  a; 

14.   — X' 


X*  —  X-  —  J, 
15.  _^^.  ^^  (a;-a)c?x 


29.  e' "dx. 
30. 


(.T-a)-  +  (.T  +  a)"'* 

16    ^'^~- —  die.  o.  {x-a)dx 

;»3  +  6a;2  +  8a;  'J-   (x- «)- ±  (a;  + a)- 

17.     — on  tZ.X- 

a^cos^aj  +  fe^sin^tc  ■^^-  x(x  +  \)- 

18.  _^:i!^.  33     ^^• 


19.  -    '^'-^    ^dx.  34.  L±idda;. 
^2mx  —  ^  1__x~- 

20.  ^  +  ^^  c?y.  35.  xWT^^dx. 
'  (^  -\-  x'  

36.  a^^i  +  x'dx. 
fy^      tndx 

aT^*  ^^-  e^'sinSajdoj. 

„„      wia;da;  38.  , 

22.  ^^:^-  a^^Vl  +  o;^ 


224  THE   INTEGRAL   CALCULUS. 


aUOCESSIVE   INTEGRATION. 

155.  Successive  differentials  obtained  on  the  hypothesis 
that  the  variable  is  equicrescent  are  readily  integrated  by  the 
preceding  methods,  the  differential  of  the  variable  being  con- 
stant. 

Examples.     1.  Given  cPi/  =  10  x"rlx-,  to  find  ?/, 


— ^  =  lOar^rfiK:  integrating,  -^ 
dx  '  *  ""  (Ix 


=  10a^dx;  integrating,  !lf  =  J^r'  +  C". 


dy  = -y- r 'd.c  +  CVZ.r  ;  integrating,  ?/ =  |a;^ -|-C".c -fC". 


d^y 
2.  Given  —4,  =  f'os  x,  to  find  y. 
dx^ 

d^y  ^  dry        .        ,  ^, 

—4  =  oos  xdx :    .  •.  — ^„  =  sin  a*  +  C  . 
doc-  dx- 

^  =  sin  xdx  +  C  'dx  ■    . :    v-  =  -  oos  x+C  'x  +C". 
dx  dx 

dy  =  —  cos  xdx  +  C  'xdx  +  C"dx  • 
.-.  y     =-sina;+^'^^--fO"x-+C"". 


d^y 
3.  Given  —4,  =  0,  to  find  y. 
dar 

2  =  0;.-.  g  =  C".     dy  =  C'dx;    ,.y  =  C'x+C". 


4.  Given  d*y  =  sin  xdx*,  to  find  y. 

5.  Given  d-s  =  —  gdf,  to  find  s. 

6.  Given  — ^  = ,  to  find  y. 

dx"         x"  ^ 


THE   CONSTANT    OF    INTEGRATION.  225 


THE   CONSTANT   OF   INTEGRATION. 

156.  All  the  integrals  thus  far  obtained  contain  the  inde- 
terminate constant  C,  and  are  called  indefinite  integrals. 

Integrals  from  which  the  constant  has  been  eliminated,  or 
for  which  its  value  has  been  determined,  are  called  definite 
integrals. 

157.  Definite  integrals.  The  two  methods  of  disposing  of 
the  constant  of  integration  G  are  best  explained  by  an  illus- 
tration of  the  processes.     Let  it  be  required 

to  find  the  plane  area  OM'N^  between  the 
parabola  0M\  the  ordinate  M'N\  and  the 
axis  of  X.  This  area  may  be  regarded  as 
generated  by  the  motion  of  the  ordinate  PD 
from  left  to  right.  If  this  area  be  repre- 
sented by  z,  dz  will  represent  what  its  change 
would  be  in  any  interval  of  time,  dt,  if  its  rate  of  increase 
remained  uniformly  the  same  during  that  interval.  But  if  the 
rate  of  z  becomes  constant  at  any  instant,  that  is,  at  any  value 
PD  of  y,  its  increase  for  any  interval  dt  will  be  represented  by 
PQRD  =  PD  X  DR  =  ydx ;  DR  =  dx  being  the  corresponding 
differential  of  x.     Hence  dz  =  ydx,  and 


=jyda 


(1) 

y 

5  value   dx  =  "-  dy  from  the  equation  of 
parabola  y-  =  2pic, 


y 

Substituting  the  value  dx  = "-  dy  from  the  equation  of  the 


dz  =  jdy  (2) 

and  z=:^i\/dy=^f-^a  (3) 

First  Method.    Evidently  the  area  generated  cannot  be  defi- 
nitely expressed  until  we  assume  some  initial  position  of  PD 


226  THE  INTEGRAL  CALCULUS. 

as  an  origin  from  which  to  estimate  it.     If   we  reckon   the 
area  from  the  ordinate  through  the  focus  F,  then  «  =  0  when 

7)' 

y=FP'=p,  and  (3)  gives  C=—\^,  and  the  definite  integral  is 

o 

which  gives  the  area,  estimated  from  FP',  to  any  position  of 
y  as  M'N'  when  y'  =  M'N'  is  substituted  for  y. 

If  we  reckon  the  area  from  0,  then  2;  =  0  when  y  =  0,  and 
(3)  gives  C  =  0,  the  definite  integral  being 

op 

which  gives  the  area,  estimated  from  0,  to  any  position  M'N' 
of  y,  when  y'  =  M'N'  is  substituted  for  y. 

Hence  the  value  of  C  may  be  found  whenever  we  know  the 
value  of  the  function  for  a  particular  value  of  the  variable  ; 
and  it  is  evident  that  this  will  be  the  case  in  all  problems  like 
the  above,  in  which  the  origin  from  which  the  magnitude  is  to 
be  estimated  may  be  arbitrarily  chosen. 

Second  Method.  If  we  substitute  any  value  of  y,  as 
y"=  M"N",  in  (3), 

z"  =  I-  4-  C 
Sp 

is  the  area  generated  while  the  ordinate  is  moving  to  the  posi- 
tion M"N".     Substituting  ?/'  =  M'N', 

z'  =  l-  \-C 
3p 

is  the  area  generated  while  the   ordinate   is   moving   to  the 
position  M'N'.     Hence 


is  the  area  generated  in  moving  from  M'N'  to  M"N",  and  is 
independent  of  any  initial  position  of  the  ordinate.     In  other 


THE   CONSTANT   OF   INTEGRATION.  227 

words,  the  area  is  increasing  at  the  rate  —  =  ^  -^,  and  the 

dt      p  dt 

area  generated  at  that  rate  while  y  passes  from  the  value  y' 

to  the  value  y"  is  found  by  substituting  these  values  in  (3) 

and  taking  the  difference  of  the  results.     In  this  way  C  is 

eliminated,  the  process  being  called  integration  between  limits. 

The  symbol  for  the  integral  between  the  limits  y'  and  y"  is 

Xy" 
(f)(y)dy,  y"  being  the  superior  and  y'  the  inferior  limit; 

and  it  indicates  that  in  the  integi-al  of  (ji(y)dy,  y"  and  y'  are 
to  be  substituted  for  the  variable  in  succession,  and  the  latter 
result  subtracted  from  the  former.  It  is  to  be  observed  that 
the  two  methods  are  essentially  the  same,  for  in  the  first  the 
inferior  limit  is  assumed  in  determining  the  value  of  C,  and 
the  superior  limit  is  the  value  subsequently  assigned  to  the 
variable  in  the  definite  integral. 

The  constants  introduced  in  successive  integration  are 
readily  determined  from  the  conditions  of  the  problem  if  the 
latter  is  a  determinate  one. 

Thus,  suppose  a  body  starts  from  rest  with  a  constant  acceler- 
ation m  in  a  right  line.  Taking  the  axis  of  X  coincident  with 
the  rectilinear  path,  we  have  (Art.  59), 

d'x 

— n-=  m. 

dt' 

Multiplying  by  dt  and  integrating, 

'I^  =:  V  =  mt  +  a  (1) 

dt 

Reckoning  t  from  the  instant  the  body  starts,  we  have,  by 
condition,  v  =  0  when  t  =  0',  hence  C=0,  and 

^  =  v=:mt.  (2) 

dt  ^  ' 


Integrating  again, 


x  =  '^  +  C\  (3) 


228  THE  INTEGRAL  CALCTLUS. 

Reckoning  x  from  the  initial  position  of  the  body,  x  =  0 
when  t  =  0;  hence  C"  =  0,  and 

mt-  , . . 

x=^.  (4) 

Eliminating  t  between  (li)  and  (4),  we  have  for  the  equa- 
tions of  motion, 

,  mt-  /n 

V  =  mt,    X  =  -— ,    V  =  V  ^  mx, 

from  which  we  may  find  the  position  of  the  body  at  any  time, 
and  its  velocity  at  any  time  or  at  any  point  of  the  path. 

Had  the  body  an  initial  velocity  Vq  when  x=t=0,  we  should 
have  had  from  (1),  C=Vo,  and  therefore 

~  =  v  =  mt-\-  Vo ; 
at 

whence,  integrating  again, 

in  which  C'  =  0,  since  x  =  0  when  ^  =  0.     The  equations  of 
motion  in  this  case  would  be 

v  =  mt  +  Vo,   X  =  -—  +  tV,   V-  =  Vq-  +  2  mx. 

And,  in  general,  the  equations  of  motion  can  be  found  when- 
ever the  position  and  velocity  of  the  body  at  any  instant  is 
known. 


CHAPTER   VIII. 

GEOMETRICAL    APPLICATIONS. 

158.   Determination  of  the  equations  of  curves. 

1.  To  find  the  equation  of  the  curve  ichose  normal  is  constant. 
Let  R  =  length  of  normal.     Then  (Art.  27,  Ex.  21), 

or  .c  =  ±fy{Ii'  -  r)-^  dy  =  ip  {H'  -  f)  -  +  C.  (1) 

la  this,  as  in  all  like  cases,  the  fact  that  the  position  of  the 
origin  of  coordinates  is  arbitrary  enables  us  to  determine  C. 
Thus  if  we  assume  that  the  origin  is  so  chosen  that  y  =  R 
when  x  =  0,  then,  from  (1),  C  =  0.  Hence  ic  =  q:  V^^  —  y^ 
or,  squaring,  a^  -\-y^  =  R^;  the  curve  being  a  circle,  and  the 
constant  of  integration  being  determined  upon  the  condition 
that  the  origin  is  at  the  centre. 

2.  To  find  the  curve  ichose  snbtangent  is  constant. 

doc 

7/  —  =  m ;  hence  x  =  log„  y  +  C,  or  a;  =  log„  y  if  x  =  0  when 
dy 

y  =  1- 

See  Ex.  8,  Art.  30. 

3.  To  find  the  curve  whose  subnormal  is  constant. 

dv 
y-^=P'     Hence  y-  —  2j)x  if  .t  =  0  when  y  =  0. 

229 


230  THE  INTEGRAL  CALCULUS. 

4.  To  find  the  curve  tohose  subnormal  is  always  equal  to  the 
abscissa  ofthej^oint  of  contact. 

An  equilateral  hyperbola. 

5.  To  find  the  curve  ichose  tangent  is  constant. 

V\  1+    —  I  —a:    whence  dx  =  x  ^^ — ~y  )    c^,/ 

Taking  the  negative  sign,  that  is,  the  case  in  which  y  is  a 
decreasing  function  of  x, 

^^_CW-y-y  Cf      g- y\ 

^          y  ^\y(a'-y')^      {a?-,/)^) 

=  a  log —^ — '-^  -  (a-  -  r )  -"  +  C.  (Ex.  5,  Art.  145. ) 


Assuming  the  origin  so  that  x  =  0  when      j- 
y  =  a,  we  have  C  =  0.     The  curve  is  called 
the  tractrix,  and  is  shown  in  the  figure. 


Fig.  72. 


6.  Find  the  curve  whose  polar  subtangent  is       0  x 
constant. 

dB 
i^  —  =  a   (Art.  120).     The  reciprocal  spiral. 
dr 

7.  Find  the  curve  whose  x>olar  subnormal  is  constant. 


159.    Rectification  of  plane  curves.     The  ])rocess  of  finding 
the  length  of  a  curve  is  called  rectification. 


I.  To  rectify  f{x,  y)  =  0.     ¥'rom  Art.  25,  ds  =  Vdaf  -\-  dy^ ; 
hence 

^s=  CVda^  +  dy\  (1) 

II.  To  rectify  f{r,  0)  =  0.    From  \rt.  120,  ds=VdJ^+i^d¥; 
hence 

s  =  C^di^  +  r^d^.  (2) 


GEOMETRICAL   APPLICATIONS.  231 

By  substituting  the  value  of  dx,  oi  of  dij,  from  the  equation 
of  the  curve  in  (1),  s  may  be  expressed  in  terms  of  a  single 
variable  and  its  value  found  when  the  integration  is  possible. 
If  the  curve  is  given  by  its  polar  equation,  the  second  form 
of  s  is  in  like  manner  expressed  in  terms  of  a  single  variable. 

Examples.     Rectify  the  following  curves  : 
1.    The  semi-cubical  parabola  y^  =  ax\ 
dy  =  ^^axdx; 

hence  .s  =  \  ("(4  +  i) ax)  ^-dx  =  -~  (4  +  0 ax) '  +  C. 

Estimating  the  length  from  the  vertex,  «  =  0  when  a;  =  0 ; 

.-.   C  =  -  — ,  and  .s  =  -^  [  (4  +  \)  ax) '  -  8], 
'11  a  '27  a 

which  is  the  length  of  the  curve  from  the  vertex  to  any  point 
whose  abscissa  is  x. 


2.   The  cycloid  x  =  r  vers"' " ,  —  V2  i^y  —  y'^. 

y 


2ry 


hence  s  =  Vl' r  (  (2  r  -  y)~'-dy  =  -  2 V^ v-(2 r  -y)^  +  C. 

Estimating  from  the  origin,  s  —  O  when  y  =  ();  whence 
C=4r, 
and  s  =  —  2  V2  r  (2  r  —  y)  -  +  4  r]^=2.  =  4  r. 

Hence  the  entire  length  of  one  branch  is  8  r. 

3.   The  parabola  y-=  'Jpx. 

«  =  1  C(j/  +  f)^.dy  =  ^V^n^+^log{y  +  ^fTp')  +  a 
pJ  Jj)  2 

(See  Art.  147,  II.,  the  illustrative  example.) 


232  THE    INTEGRAL    CALCULUS. 

Estimating  from  the  vertex, 


4.    The  catenary  7/  =  -  (e''  -j-  e  '^). 

Estimating  the  arc  from  the  point  for  which  x  =  0, 

z  z 

C 


=:,(«'-«  0- 


5.  The  hypocycloid  a;' -f-//^  =  a''.     Ans.  6a. 

6.  Determine  tlie  length  of  the  tractrix. 
From  Ex.  5,  Art.  158,  ^ 

dx-  =  — ^^^  dy'^. 

y- 

Hence  s=  I  Vd-ir'  +  dy-  =  —  |  ?(/^  =  —  a  kig y  -}-  C, 

taking  the  negative  sign  as  s  is  a  decreasing  function  of  y. 
(Fig.  72.)  Estimating  the  arc  from  T,  s  =  ()  when  y  =  a; 
hence  C=aloga-,  and  s  =  alog-. 

y 

7.  Determine  the  length  of  the  ellipse. 

Using  the  central  form  of  the  equation  in  terms  of  the 
eccentricity, 

2/^  =  (l-.^)(a^_a-),  c^^^^iLziir)^; 

(r  —  X- 
hence 

s  =  CVdx"  +  dy'  =  fxrr  ^'f  da;  =  f — ^^-      (a-  -  e'.v')  ^- ; 
J  J   \  a'-x-  J  Va-  -  x" 

and  for  the  length  of  the  entire  curve. 


GEOMETRICAL   APPLICATIONS.  233 

3eV 


1  =  41     — =^zz=r  I  « „ 

Jo  v;?3^V        -'«      2.4a-^      2. 


4  •  6  a' 

(Ex.  25,  Art.  72.) 


~      Jo  Vrt*  -  y?     "'«^''  Va'-.T^      2  a'' Jo  Vo^^ 


„      ..       e^      3e*        32 
27ra(l---  — - 


22     2^2     2^ .  42 .  6- 

The  second  and  third  of  the  above  integrals  are  given  in 
Art.  147,  Exs.  1  and  2. 

8.    The  logarithmic  spiral   r  =  a*,  a  being  the  basis  and  m 
the  modulus  of  the  logarithmic  system. 


dr  =  -de ;  s  =  C(^^  +  aA^^dO  =  (1  +  m')^ r  +  cT=  Vl  + 
m  J  \m-  )  Jo 

the  length  from  the  point  for  which  r  =  1  to  the  pole. 
The  corresponding  arc  of  the  Naperian  spiral  =  V2. 

9.    The  spiral  of  Archimedes,  r  =  aQ. 

s  =  a  C^T+J'dO  =  i  C{a'^  ?-') ^dr 


nr 


^r(a^-^,^r_^a       r  +  ^a^  +  r-^    (Art.  147,  11.) 
2a  2  a 

when  the  arc  is  estimated  from  the  pole.  This  is  also  the 
length  of  the  arc  of  the  parabola  y^  =  2ax  from  the  vertex  to 
y  =  r  (Ex.  3) ;  hence  this  spiral  is  often  called  the  parabolic 
spiral. 

10.     r  =  a(l  +  cos^). 

S  =  C(di^  -f-  ^-^dO'-)  ^  =  C^2d'{l-\-cosd)de 
=  r^4a2cos''^c7^  =  2a('cosfdd  =  4a  sin^  +  C. 


234  THE  INTEGRAL  CALCULUS. 

Estimating   the   area  from  the  point  for  which  0  =  0,  we 

a 

have  C=0,  and  s  =  4asin  — 

2 

The  curve  is  a  cardioide,  the  polar  axis  being  the  axis  of 
symmetry,  and  its  entire  length  is  8  a. 

160.   Quadrature  of  plane  areas.    I.  The  plane  area  included 
between  y  =f{x)  and  the  axis  of  X  is  given  (Art.  157)  by 

2  =jydx.  (1) 

In  like  naanner  z'  =  j  xdy  gives  the  area  between  the  curve 

and  Y. 

dz 
If  the  curve  crosses  X,  y,  and  therefore  — ,  becomes  nega- 

dx 

tive,  z  being  a  decreasing  function  of  x ;  hence  areas  below  X 
must  be  considered  as  negative. 

II.   By  the  area  of  a  polar  curve  is  meant  the  area  swept 
over  by  its  radius  vector.     Thus  OPQ  is  the 
area  of  MN  between  the  limits  P  and  Q. 

Eepresenting  the  area  by  z,  its  change  would 
evidently  become  uniform  at  any  value  of 
r=  OP  if  at  this  value  the  generating  point 
moved  uniformly  in  the  circular  arc  PP'. 
Hence  if  d$  =  j)p', 

dz  =  area  OPP'  =  ^OPx  PP'  =  ir-  rdO, 

or  2  =  1  Cj-^rie.  (2) 

The  process  of  finding  the  area  is  called  ftuadrature. 

Examples.     1.  Determine  the  area  of  the  parabola  y'^=2px. 
dx  =  tdy;  hence 


z=  \  ydx  =      i  yhly  =  -^  +  (7. 


GEOMETRICAL   APPLICATIONS. 


235 


Estimating  the  area  from  the  vertex,  z  =  0  when  y  =  0  ;  hence 
C=0,  and  2;  =  ~-  =  |a^,  or  two-thirds  the  circumscribing 
rectangle. 

2.  Determine  the  area  between  y  =  s^in  x  and  X. 

z  =  i    sin  xrlx  =  —  cos  x 

3.  Show  that  the  area  between  the  witch  x-y  =  4'/--(2/-  —  y) 
and  its  asymptote  is  47r)-. 

ST^dx 


fydx=f 


=  4?-^tan~'  — 
a^+4?"  'Jr 


=  27rr2-(-27r»-2)  =  47rr^. 


4.  Show  that  the  area  between  X  and  the  hyperbola  xy  =  1 
from  X  =  1  to  X  =  .t'  is  log  a;'. 

5.  Find  the  area  of  one  branch  of  the  cycloid. ' 
Cyrix  =  C-y'^y       =Cr{2r-y)-^dy 

«/  ./      -v/'>  n-tl   7/2  »/ 


V2  ry  —  ?/ 
=  —  -^—^ (2  ?•?/  —  ?/)  -  -I vers  '  •- 


=  f7rr^; 


2       ■  :^  ?• 

hence  the  whole  area  is  .StD"'.     See  Ex.  9,  Art.  147. 

6.  Find  the  area  of  the  circle  x-  +  y^  =  ?-. 
fydx=f(^^-x^)kx  =  ^(^-^  +  ^^si.'^^^^^ 

hence  the  whole  area  is  tt?*^.     See  Ex.  14,  Art.  147. 

7.  Prove  that  the  area  of  the  ellipse  a^y-  -f  b^a^  =  a'b^  is  irab. 

8.  Show  that  the  area  between  the  cycloid 

y  

x  =  2  vers~^  |  —  V4  y  —  y'^ 

c 

and  the  parabola  y-=  -a;  is  |7r. 

The  curves  intersect  at  the  origin  and  x  =  2ir. 


286  THE  INTEGRAL  CALCULUS. 

9.  Find  the  area  of  y'^  =  a;*  +  ar'  on  the  left  of  Y  (see  Fig.  57). 

fydx  =  Cf{l  +  X) 2dr  =  2  Cz\z'  -  Ifdz 

See  Ex.  7,  Art.  143. 

10.  Show  that  the  area  of  the  loop  a^y*  =  a-x*  —  af  is  ^o.-. 

11.  Show  that  the  area  of  y(af  -f  a^)  =  c^(a  —  x)  from  a;  =  0 
to  a;  =  a  is  c^(^\og2  —  -y 

x^ 

12.  Prove  that  the  area  between  the  cissoid  y^  = and 

"^a  —  x 
its  asymptote  is  Stto^.     See  Ex.  9,  Art.  147. 

13.  Prove  that  the  area  of  both  loops  of  y'^  =  xr{l  —  x^y  is  |-. 
See  Fig.  59. 

14.  Prove  that  the  area  between  X  and  y  =  4x  —  a:^  from 
a;  =  —  2  to  a;  =  +  2  is  8. 

15.  The  spiral  of  Archimedes,  r  =  aO. 

z  =  i  Circle  =  ~  Cme  =  -&'  =  ^  r'e, 

C  being  zero  if  the  area  is  estimated  from  ^  =  0.  For 
^  =  2 TT,  z=  ^ irr^,  or  the  area  of  the  first  spire  is  ^  that  of  the 
measuring  circle.  When  6  =  4:7r,  z  =  ^Tr'i'^,  or  the  area  of  the 
second  spire  is  firr^  —  |7rr^  =  27r7-^,  the  first  spire  having  been 
traced  twice. 

16.  Prove  that  the  area  of  r  —  e^  is  one-fourth  the  square 
described  on  the  radius  vector. 

17.  Find  the  area  of  the  lemniscate  7-^  =  a^  cos  2  d.      Ans.  a*. 

18.  Prove  that  the  area  of  the  cardioide  7-  =  a(l+cosd) 
is  f  iral 


GEOMETRICAL   APPLICATIONS. 


237 


19.  Prove  that  the  area  of  the  three  loops  of  r  =  a  sin  3  0 
(Fig.  67)  is  ^Tra'. 

20.  Find  the  area  of  the  four  loops  of  r=  asin2^  (Fig.  66). 

161.   Volumes  and  surfaces  of  revolution. 

Let  the  curve  ON,  whose  equation  is  y  =f{x),  revolve  about 
X  as  an  axis  of  revolution.  The  plane  area  OQR  will  generate 
a  solid  revolution  whose  surface  will  be  gener- 
ated by  OQ.  A  plane  section  PP  perpendicu- 
lar to  X  will  cut  from  this  solid  a  circle  whose 
centre  is  D  and  radius  is  PD  =  y.  The  volume 
of  the  solid  may  be  regarded  as  generated  by 
this  variable  circle  moving  with  its  centre  on 
X.     The  rate  of  every  point  of  this  generating 

area  is  — ;  hence  the  rate  of  increase  of  the 
dt 

,  .    dV         odx 

volume  V  is  —  =  Trir  — ,  or 
dt        ^  dt 


V=  i  iryHx  =  TT I  y^c^a 


(1) 


The  surface  S  of  the  solid  may  be  regarded  as  generated  by 
the  circumference  of  the  circle.     The  rate  of  every  point  of 

this  generating  circumference  is  — ;  hence  the  rate  of  increase 

of  the. surface  is  —  =  2Try—,  or 
dt  dt 


S=  C2  tryds  =  2  TT  Cy^dx"  +  dy\ 


(2) 


Examples.     1.  Find  the  volume  of  the  paraboloid  of  revo- 
lution. 

V=ir  \  ifdx  =  Tr  I  2pxdx  =  irpsc^  -\-C.     Estimating  the  vol- 
ume from  the  vertex,  F=  0  when  x  =  0;   hence  C  =  0,  and 

V  =  irpx^  =  irpx  2^  =  \  -rrfx, 
or  one  half  the  volume  of  the  circumscribing  cylinder. 


238  THE   INTEGRAL   CALCULUS. 

2.  Find  the  volume  of  the  prolate  speroid. 

Jr'a  J2 
—  (a?  —  oi?\dx  =  ^TTh-a.     Hence  the  whole  volume 
"    a- 

=  |7r6-(2a),   or  two-thirds  the  circumscribing  cylinder.      If 

.3.  Find  the  volume  of  the  oblate  spheroid. 
Here  F=7r  Cx'dy  =  ^7ra^{2b). 

4.  Find  the  volume  generated  by  the  revolution  oiy—  —  x-\-h 
about  X:  " 

/r  I    y'hlx  =  ^  Trb'-a. 

5.  Find  the  volume  generated  by  the  revolution  of  the  cycloid 
about  X 

/TTi/dx  =  I  -n-y^  —  y  y      ~  '^  I  .^'(-^ ''y  ~  V'Y^y 
^        V2  ry  —  y'        ^ 

2  y-  +  r>r(y  +  3  r)    /.^ t, 

=  —  TT  —^ — f^—^ i  V2  ry  —  y' 

y 
+  #7r?'"  vers' ' "  -f-  C 

or  the  whole  volume  =  5  ttV.     See  Ex.  15,  Art.  147. 

6.  Find  the  volume  generated  by  the  revolution  of  the  witch 
about  its  asymptote,     x^y  =  4?-^(2r  —  ?/)  ; 

TT  CyHx  =  TT  f— Mr!_^  dx  =  64  /tt  f ^^— 

=  64  rV  (^——4 7-  +  -^  tan-i  — ^ 

\^8?'2(47-2  +  ar')      IBr*  2ry 


See  Ex.  12,  Art.  147, 


=  47rV. 


7.  Show  that  the  volume  generated  by  the  revolution  of 
a.5  _|_yt  _  ^t  about  the  axis  of  X  is  y^^Tra^ 


GEOMETRICAL   APPLICATIONS.  239 

8.  Find  the  surface  of  the  paraboloid  of  revohition. 

S  =  27r JyVdx'  +  clf'=2^JyyJy^^  +  lrh,  =  ^^^y'+p'y^+C. 

Estimating  the  surface  from  the  vertex,  S  =  0  when  y  =  0; 

whence  C=  — --x>•^  and  S  =  -^[(y- +p'^)^ —p'^'l. 
3p  3p^^  '  -• 

9.  Find  the  surface  of  the  sphere. 

.S  =  2 TT  (^'^/-J^  +  1  rfx-  =  2 TT  rV  sf-^—- dx=27r  Crdx  =  4 


TTV. 


10.  Find  the  surface  generated  by  the  revokition  of  x^  -\-y^=a^ 
about  X.  Ans.  ^-tra?. 

11.  Find  the  surface   generated   by  the   revolution   of  the 
cycloid  about  its  base. 

S  =  2ir  ryy\~^^-^  +  \dy  =  2n-y/2^'  ry{2r -yyhy 
Jo       ^2ry  —  y^  Jo 

=  -2W27-(|(4r+^)(2r-i/)2)]2'=^2^r^. 
Hence  the  whole  surface  =  -%^  irr-.     See  Ex.  5,  143. 

12.  Prove  that  the  surface  generated  by  the  revolution  of 
one  branch  of  the  tractrix  about  X  is  27ra^.    See  Ex.  5,  Art.  158. 

13.  Prove  the  area  of  the  surface  of  the  prolate  spheroid  is 

tto.-] — sin  'e. 


1  14  DAY  USE 

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